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THE  INDUCTION  MOTOR 


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Published  by  the 


Me  G  raw  -  H  ill   B  ook.  Comp  any 

' 


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Publishers   of  Books  for 

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THE 


INDUCTION    MOTOR 


BY 


BENJ.   F.  BAILEY,  B.S.,  Pn.D. 

Junior  Professor  of  Electrical  Engineering;    University  of  Michigan 


LIBRARY 

Southern  Cdifonifa  Edison  Company 


McGRAW-HILL    BOOK    COMPANY 

239   WEST   39TH   STREET,   NEW   YORK 

6  BOUVERIE  STREET,   LONDON,   E.G. 

IQII 


COPYRIGHT,  1911, 

BY 
McGRAW-HILL  BOOK  COMPANY 


itical 

3 

Library 


PREFACE 


IN  presenting  a  new  treatise  on  the  induction  motor,  the  writer 
is  aware  that  he  is  entering  a  field  in  which  there  are  already  many 
excellent  works.  In  this  book,  however,  an  attempt  is  made  to  pre- 
sent the  subject  from  a  somewhat  new  standpoint.  The  endeavor 
has  been  to  produce  a  work  that  will  have  the  greatest  pos- 
sible value  for  those  who  wish  to  inform  themselves  more  fully 
regarding  the  theory  of  the  induction  motor  than  they  can  by  studying 
the  elementary  text-books,  but  who  at  the  same  time  do  not  car'e 
to  go  too  deeply  into  the  theoretical  aspects  of  the  question.  The 
writer's  aim  has,  therefore,  been  to  present  so  much  of  the  theory  as  is 
necessary  to  understand  the  phenomena  of  the  induction  motor,  so  far 
as  these  phenomena  relate  to  the  design  or  operation  of  these  machines. 

The  student  is  assumed  to  have  some  knowledge  of  alternating 
currents,  and  to  understand  in  a  general  way  the  operation  of  the 
alternator,  the  synchronous  motor,  the  induction  motor,  etc.  To 
this  end,  he  is  supposed  to  have  read  some  of  the  several  elementary 
texts  dealing  with  these  subjects. 

Throughout  the  book,  an  earnest  endeavor  has  been  made  to 
present  clearly  the  physical  conception  of  the  actions  taking  place. 
It  is  the  writer's  belief  that  nine-tenths  of  the  trouble  experienced  by 
many  people  in  studying  the  action  of  electrical  machinery  comes 
from  the  lack  of  a  clear  idea  of  the  elementary  physical  actions.  An 
attempt  is  made  to  apply  mathematical  reasoning  to  the  problem 
before  this  understanding  is  obtained,  and  the  result  is  a  mental 
haze,  which  is  perhaps  never  dissipated.  The  reader  is  therefore 
strongly  urged  to  study  carefully  the  first  two  chapters,  and  make 
sure  that  they  are  fully  understood,  before  going  farther. 

Several  subjects  of  the  greatest  practical  importance  have  been 
only  briefly  mentioned,  if  treated  at  all,  by  previous  writers.  Some 
of  these  are  the  variation  of  the  starting  torque  in  different  positions 
of  a  wound  rotor,  the  disadvantage  of  too  great  starting  torque  in 
squirrel-cage  motors,  the  iron  losses  in  the  rotor  teeth,  etc.  It  is 
also  thought  that  the  examples  of  design  given  in  most  of  the  books 


vi  PREFACE 

on  this  subject  are  not  representative  of  present  conditions.  This 
is  on  account  of  the  fact,  well  known  to  designers,  that  in  recent  years 
a  marked  reduction  in  the  weight  and  dimensions  of  induction  motors 
has  taken  place.  It  is  believed  that  the  examples  of  design  given 
are  fairly  representative  of  average  modern  motors. 

The  writer  washes  to  take  this  opportunity  to  acknowledge  his 
indebtedness  to  those  who  have  preceded  him  in  this  field.  Every 
work  of  an  engineering  nature  is  necessarily  founded  on  that  of 
others.  An  attempt  to  give  original  demonstrations  of  all  of  the 
elementary  facts  in  relation  to  a  subject,  necessarily  leads  to  a  far 
less  simple  treatment  than  the  frank  use  of  older  methods,  when  these 
methods  are  at  least  as  clear  as  anything  the  author  has  to  offer  in 
their  stead.  In  this  regard,  the  present  treatise  is  no  exception. 
Particular  mention  should  be  made  of  the  excellent  works  of  Behrend, 
Boy  de  la  Tour,  and  McAllister.  In  particular,  the  treatise  of 
the  last  named  author  has  been  drawn  upon  for  several  simple  and 
lucid  demonstrations,  notably  for  the  proof  of  the  circle  diagram, 
and  for  one  of  the  methods  of  treating  the  subject  of  single-phase 
motors.  Credit  has  been  given  in  the  text  for  contributions  of 
various  writers.  When  this  has  not  been  done,  the  omission  is  due 
either  to  a  lack  of  knowledge  on  the  part  of  the  author,  of  the  work 
in  question,  or  an  uncertainty  regarding  the  author  to  whom  such 
credit  should  be  given. 

Lack  of  time  and  space  has  caused  the  omission  of  much  mate- 
rial of  historical  interest.  The  same  reason  has  caused  the  omission 
of  the  discussion  of  many  ingenious  devices  designed  to  improve  in 
various  ways  the  performance  of  induction  motors.  Many  of  these 
relate  to  methods  of  improving  the  starting  performance  of  induction 
motors,  and  present  numerous  points  of  theoretical  and  practical 
importance.  In  order  to  keep  the  size  of  the  book  within  reasonable 
limits,  it  was  thought  best  to  limit  the  discussion  almost  entirely 
to  devices  in  successful  operation,  and  to  those  of  recent  introduction 
which  seem  most  likely  to  be  of  permanent  value.  For  the  same 
reason,  the  treatment  has  been  almost  entirely  confined  to  motors 
built  in  the  United  States. 

For  much  of  the  experimental  work  mentioned  in  the  text  the 
author  is  indebted  to  the  careful  work  of  Mr.  H.  L.  Tanner,  Instructor 
in  the  Department  of  Electrical  Engineering  of  the  University  of 
Michigan,  and  to  Mr.  Stanley  D.  Livingstone,  assistant  in  the  same. 

ANN  ARBOR,  MICH.,  September,  1911. 


TABLE   OF   CONTENTS 


PAGE 

PREFACE  v 

CHAPTER  I.— ELEMENTARY  THEORY i 

Definitions  of  stator  and  rotor — Simple  description  of  windings — Expla- 
nation of  rotating  magnetic  field — Synchronous  motor  action— How 
current  is  induced  in  the  rotor — Variation  in  rotor  e.m.f.  with  slip — 
Lag  of  the  current  in  the  rotor — Shape  of  flux  wave — Influence  of  the 
rotor  current  on  the  flux — Vector  relations  of  current  and  e.m.f. — 
Relation  of  induced  e.m.f.  and  flux— Magnetizing  current — Relation  of 
flux  and  current — Vector  diagram,  no  load — Vector  diagram,  full  load — 
Vector  diagram  of  starting  condition — Torque — Reasons  for  using  wound 
rotor. 

CHAPTER  II.— THEORY  OF  THE  INDUCTION  MOTOR 16 

Equations  connecting  slip,  rotor  current,  etc. — Induction  motor,  equiva- 
lent to  a  transformer — Proof  of  circle  diagram — Use  of  circle  diagram  in 
testing,  etc. — Determination  of  losses,  slip,  starting  torque,  etc. — Slip 
and  torque — Relation  between  synchronous  watts  and  torque — Maximum 
value  of  power  factor— Exact  construction  of  circle  diagram. 

CHAPTER  III.— STARTING  TORQUE 30 

Proof  that  starting  torque  equals  synchronous  watts — Proof  that  this 
applies  to  any  type  of  motor — Comparison  of  starting  torque  of  various 
motors — Starting  torques  of  squirrel-cage  and  wound-rotor  machines — 
Starting  torques  of  commercial  induction  motors — Importance  of  starting 
torque  and  losses — Torque  in  terms  of  slip,  and  ratio  of  starting  to 
running  current — Undesirability  of  making  starting  current  too  large — 
Comparison  of  25  and  50  cycle  machines — Importance  of  reducing  the 
slip — Comparison  of  wound-rotor  and  squirrel-cage  machines — Influence 
of  poorer  leakage  coefficient— Effect  of  iron  losses— Fluctuation  of  torque 
in  wound-rotor  machines. 

CHAPTER  IV.— STARTING  DEVICES 41 

Secondary  starters — Direct  connection  to  line — Auto  transformers — 
Resistance  starters — Use  of  taps  on  supply  transformers — Star  and  delta 
connections  for  starting — Use  of  extra  coils  on  motor — Starting  several 
motors  from  one  starter— Comparison  of  resistance  starters  and  auto 
starters — Mathematical  proof  of  relative  line  disturbance — Carbon-block 
starters— Protective  devices. 

vii 


viii  TABLE  OF  CONTENTS 

PAGE 

CHAPTER  V. — THE  INDUCTION  GENERATOR 60 

General  considerations,  circle  diagram — Outside  excitation  generally 
necessary — Excitation  by  synchronous  motor — Excitation  by  condenser 
— Excitation  by  synchronous  generator— Use  in  connection  with  small 
water  powers — Use  in  connection  with  steam  turbines — Example  of  use 
with  synchronous  generators — Phase  converter — Changing  from  two  to 
three  phase — Starting  single-phase  motors — As  voltage  balancers. 

CHAPTER  VI. — VARIABLE  SPEED  INDUCTION  MOTORS 72 

Use  of  rotor  resistance — Control  devices — Cascade  connection :  Efficiency, 
Output,  Power  factor — Variable  of  number  poles — Commutator  type 
induction  motor:  General  theory,  Variable  speed  by  changing  secondary 
voltage,  Raising  power  factor,  Relation  of  inductance  and  speed  of 
armature,  Commutation. 

CHAPTER  VII.— MORE  EXTENDED  THEORY  OF  THE  INDUCTION  MOTOR      .     96 
Relation  of  e.m.f .  and  current — Damping  action  of  rotor — Sine  wave  of  flux 
on  account  of  rotor  action — Formula  for  e.m.f.  and  flux — Short  pitch  wind- 
ings and  formulae — The  magnetizing  current. 

CHAPTER  VIII.— LEAKAGE  COEFFICIENT       108 

Predetermination  of  circle  diagram — Leakage  fluxes — Locked  current 
from  inductance — Determination  of  "a" — Hobart's  latest  method — 
Adams'  method — Behn-Eschenburg's  method — Behrend's  method — 
Hobart's  modification — McAllister's  equations — Nature  of  "C". 

CHAPTER  IX. — GENERAL  CONSIDERATIONS  RELATING  TO  DESIGN  .  .  .123 
High-  or  low-frequency  motors — Open  or  closed  slots — Best  diameter  of 
rotor:  For  economy  of  iron;  For  economy  of  copper;  For  best  power 
factor — Length  of  air  gap — Size  of  conductors — Two-phase  vs.  three- 
phase  motors — Copper  losses  of  stator — Iron  losses  of  stator — Iron  losses 
of  rotor — Estimation  of  heating:  By  flux  density  and  current  density; 
By  "barrel"  surface;  By  core  surface;  From  tests  of  frame. 

CHAPTER  X. — FRACTIONAL  PITCH  WINDINGS 145 

Stator  and  rotor  windings:  Star  and  delta;  Pole  connections;  Full  over- 
lapping windings;  Concentric  windings. 

CHAPTER     XI.— DESIGN    OF  40-H.P.,    750-R.p.M.,    3-PnASE,    25-CvcLE, 

440-VoLT  SQUIRREL-CAGE  INDUCTION  MOTOR     .' 156 

General  considerations — Output  coefficient — Relation  of  pole  pitch  to 
core  length — Pitch  of  coils— Value  of  "C" — Conductors  per  phase — 
Size  of  stator  wire — Flux  densities  in  gap,  teeth,  etc. — Iron  loss — Current 
per  rotor  bar — Current  per  rotor  ring — Size  of  bars  and  rings  and  losses 
in  same — Magnetizing  current — Starting  torque,  Heating,  Overload 
capacity. 


TABLE  OF  CONTENTS  ix 

PAGE 

CHAPTER  XII.— SPECIAL  TYPES  OF  MOTORS 167 

General  Electric,  Type  L  motor — Wagner  type  BW  motor — Burke 
Squirrel-cage  motor — Squirrel  cage,  crane  motors — Mill  type  motors 
— General  Electric,  Type  MI  motors— Westinghouse,  Type  MS  motor — 
Typical  construction  of  various  parts — Frames:  Ordinary  type  Skeleton 
frame;  Riveted  frame;  Heavy  duty  type;  Pedestal  type;  Completely 
enclosed  frame — Rotors:  Westinghouse,  Fairbanks-Morse,  General 
Electric,  Crocker-Wheeler,  Westinghouse — General  notes  on  the  selection 
of  motors— Speeds  of  motors— Frequency— Capacity  of  motors— Squirrel- 
cage  vs.  wound-rotor  machines — Group  or  individual  drive — Motors  for 
various  applications. 

CHAPTER  XIII.— SINGLE-PHASE  MOTORS 104 

Elementary  facts — Distribution  of  flux — Nature  of  rotor  current — 
Modifications  introduced  by  the  windings  of  actual  machines — Resolu- 
tion of  rotor  currents — Simple  commutator  type,  Single-phase  motor 
— Resolution  of  fluxes — Generation  and  value  of  the  two  fluxes — Pro- 
duction of  torque — Relation  of  slip  and  losses — Relation  of  torque  and 
speed — Use  of  secondary  resistance. 

CHAPTER    XIV. — SINGLE-PHASE  COMMUTATOR  TYPE  MOTORS        ....  208 
Starting   torque — Compensation   for  power  factor — Curves   of   General 
Electric  type  RI  motor — Speed  control — By  "armature"  control — By 
"field"  control — Relation  of  speed  and  e.m.f. — Wagner  type  BK  motor. 


THE  INDUCTION  MOTOR 


CHAPTER  I 

ELEMENTARY   THEORY 

AN  induction  motor  consists  essentially  of  two  parts — a  stationary 
part,  commonly  called  the  stator,  and  a  rotating  part,  called  the  rotor. 
The  terms  primary  and  secondary  are  sometimes  used  instead  of  the 
above.  These  expressions  are  not  strictly  interchangeable,  since  the  term 
primary  refers  to  the  part  to  which  the  current  is  supplied,  and  the 


FIG.  i. — Fairbanks  Morse  Induction  Motor. 

term  secondary  to  the  part  in  which  the  current  is  induced.  Usually 
the  stator  is  the  primary,  but  in  some  cases  this  is  reversed  and  the  rotor 
becomes  the  primary.  Some  writers  use  the  terms  field  and  armature, 
from  analogy  with  direct-current  practice.  This,  however,  is  open  to 
serious  objection,  since,  as  will  appear  presently,  the  primary  is  the  part 
which  corresponds  in  function  to  the  armature  of  a  direct- current 
machine,  so  that  what  is  thus  called  the  field  is  in  reality  the  armature. 


2  THE  INDUCTION  MOTOR 

Figs,  i  and  2  show  respectively  a  complete  squirrel-cage  induction 
motor,  and  the  same  motor  unassembled.     The  active  iron  of  both  the 


FIG.  2. — Unassembled  Parts  of  Fairbanks  Morse  Induction  Motor. 

stator  and  the  rotor  consists  of  thin  laminations,  provided  with  slots 
around  the  periphery,  and  held  together  by  suitable  means  to  form  a 
rigid  structure.  Fig.  3  is  a  drawing  of  a  pair  of  typical  stator  and  rotor 


Stator  48  Slots 
Slot  Pitch  1.113 


Rotor  110  Slots 
Slot  Pitch  0.487 


Mam.  10 
FIG.  3. — Stator  and  Rotor  Laminations. 

laminations.  The  windings  of  the  stator  and  rotor  are  placed  in  the 
slots  of  the  core.  In  many  cases  the  winding  of  the  rotor  consists  of 
copper  bars  placed  in  the  slots,  and  short  circuited  at  both  ends  by  heavy 
copper  or  brass  rings. 

The  stator  and  rotor  coils  are  connected  in  the  same  manner  as 


ELEMENTARY  THEORY  3 

though  the  machine  were  to  be  used  as  an  alternating-current  generator. 
Any  of  the  well-known  windings  which  can  be  used  for  an  alternator  can 
be  employed  in  the  induction  motor.  A  few  points  of  difference  will  be 
brought  out  later.  For  the  present,  it  is  sufficient  to  point  out  that  the 
use  of  short,  or  fractional-pitch  windings,  is  more  common  in  the  case  of 
induction  motors  than  in  the  case  of  alternators. 

If  two-phase  or  three-phase  current  be  supplied  to  a  stator  properly 
wound  for  the  corresponding  number  of  phases,  a  rotating  magnetic  field 
will  be  set  up.  This  field  will  be  nearly  uniform,  and  will  rotate  with 
practically  uniform  velocity.  Its  speed  will  be  such  that  it  will  move  the 
distance  between  two  poles  of  the  stator  in  the  time  required  for  the 
current  to  complete  one-half  cycle.  This  flux,  cutting  through  the  con- 


FIG.  4. — Three-phase  Current  Waves. 

ductors  of  the  rotor,  produces  currents  in  them.  These  currents  are 
acted  upon  by  the  stator  magnetism,  and  a  torque  is  set  up  which  causes 
the  rotor  to  revolve. 

The  following  will  explain  how  this  takes  place  in  a  three-phase 
winding.  In  Fig.  4  are  shown  the  three  currents  of  a  three-phase  circuit. 
They  differ  in  phase,  as  shown  by  one-third  of  a  cycle,  or  120  electrical 
degrees.  Fig.  5  shows  a  section  of  the  stator  and  rotor  of  an  induction 
motor.  Only  about  two  poles  are  shown,  but  it  will  be  understood  that 
the  motor  might  have  any  number  of  poles.  The  number  of  slots  in  this 
case  is  taken  as  six  per  pole  or  two  per  phase  per  pole.  We  might  use 
any  number  from  one  up,  although  less  than  two  or  more  than  eight  are 
rarely  employed.  The  winding  is  supposed  to  be  a  full -pitch  winding, 
that  is,  the  coils  span  a  complete  pole  pitch,  or  from  slot  i  to  7,  2  to  9, 
etc.  For  our  present  purpose  the  end  connections  are  immaterial.  The 
only  requirement  is  that  the  coils  be  so  connected  that  those  lying  in  the 
slots  marked  A  are  all  connected  in  one  circuit  and  have  the  current  A  in 


THE  INDUCTION  MOTOR 


them.  Likewise  those  coils,  in  slots  B  are  connected  in  the  B  circuit,  and 
those  in  C  in  the  C  circuit.  It  should  not  be  inferred  that  a  slot  always 
has  current  of  only  one  phase  in  it.  It  is  always  true  when  we  employ 
full-pitch  windings,  but  is  not  the  case  with  short -pitch  windings. 

Consider  the  currents  in  the  stator  at  the  time  marked  i  on  Fig.  4. 


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FIG.  5. — Distribution  of  Currents  in  Stator  of  a  Three-phase  Induction  Motor. 

The  current  A  may  be  regarded  as  negative,  and  both  B  and  C  as  posi- 
tive. The  relative  values  of  these  currents  are  represented  below  the 
slots  of  Fig.  5  by  arrows,  whose  lengths  are  proportional  to  the  strength 
of  the  current  and  whose  directions  correspond  to  the  direction  of  the 
current. 

Between  the  slots  B  and  C  will  be  seen  a  waved  arrow.  This  indi- 
cates the  point  of  greatest  magnetic  flux,  and  its  direction  shows  the 
direction  of  the  flux.  The  waved  arrows  then  show  the  centers  of  the 
poles  of  the  stator  at  any  particular  instant.  The  maximum  flux  is 


ELEMENTARY  THEORY  5 

obviously  located  at  the  point  shown  by  the  arrows,  since  all  the  currents 
to  the  left  of  it  are  in  one  direction,  and  all  those  to  the  right  are  in  the 
opposite  direction. 

At  the  time  marked  2  in  Fig.  4,  the  current  A  has  decreased  some- 
what, while  C  has  increased,  and  B  has  dropped  to  zero.  Both  A  and  C 
retain  the  same  direction.  The  values  of  the  currents  in  the  various 
slots  are  as  shown  by  the  arrows  opposite  the  number  2.  The  position 
of  the  maximum  flux  is  obviously  in  the  position  indicated. 

Likewise,  the  state  of  the  flux  at  the  times  3,  4,  5,  etc.,  are  as  shown 
opposite  the  numbers  3,  4,  5.  It  will  be  seen  that  the  arrow  indicating 
the  position  of  maximum  flux  moves  steadily  to  the  left.  Each  incre- 
ment of  time  considered  is  that  corresponding  to  30  degrees.  Since 
there  are  twelve  slots  per  pair  of  poles,  and  since  the  flux  moves  one  slot 


FIG.  6.— Star  and  Delta  Coil  Connections. 

for  each  change  of  current,  it  is  evident  that  it  moves  30  electrical  degrees 
for  each  change  of  30  degrees  in  the  current.  Consequently,  the  flux 
revolves  in  synchronism  with  the  current,  or  it  moves  twice  the  pole 
pitch  for  each  complete  cycle  of  current. 

We  could  in  the  same  way  show  that  if  the  stator  were  wound  for 
two  phases  instead  of  for  three,  and  if  two-phase  currents  were  supplied 
to  it,  a  rotating  magnetic  field  would  be  set  up  as  before.  This  the  reader 
can  readily  verify  for  himself. 

It  will  also  be  apparent  that  if  the  direction  of  the  currents  in  two 
of  the  windings  be  reversed,  the  direction  of  rotation  of  the  field,  and 
consequently  that  of  the  rotor,  will  be  reversed.  This  is  done  in  the  case 
of  the  actual  motor  by  simply  reversing  the  connections  of  two  of  the 
three  leads.  This  is  possible,  since  in  practice  the  three  windings  of  the 
three  phases  are  always  connected  together  in  a  Y  or  a  delta  connection, 
as  shown  in  Fig.  6.  Thus  three  wires  only,  instead  of  six,  are  required 
to  supply  current  to  the  motor. 


THE  INDUCTION  MOTOR 


PRODUCTION  OF  CURRENT  IN  THE  ROTOR 

If  in  a  stator  such  as  that  just  described  we  place  the  revolving  field 
of  an  alternator,  the  machine  will  in  all  essentials  be  an  alternating-cur- 
rent generator  and  may  be  operated  as  such.  Suppose,  however,  that 
we  supply  current  to  this  stator  so  that  a  rotating  magnetic  flux  is  set 
up  in  the  way  just  described,  and  let  us  drive  the  field  by  applying  power 
through  the  shaft  at  exactly  the  same  speed  as  that  of  the  rotating  mag- 
netic flux  due  to  the  stator  current.  If  now  the  driving  force  be  removed, 
it  is  evident  that  the  poles  of  the  rotating  field  will  be  attracted  by  the 
magnetism  set  up  by  the  stator  current,  and  will  be  dragged  around  by 
them.  The  field  will  then  continue  to  revolve  at  synchronous  speed, 


FIG.  7. — Distribution  of  Currents  in  Rotor  of  an  Induction  Motor. 

and  will  maintain  the  same  speed  irrespective  of  the  load  applied,  pro- 
viding the  torque  is  not  so  great  as  to  cause  the  poles  to  pull  entirely  away. 
Such  an  arrangement  constitutes  a  synchronous  motor.  The  action  in 
the  case  of  an  induction  motor  is  similar,  except  that  the  current  to  mag- 
netize the  rotor,  instead  of  being  supplied  in  the  shape  of  direct  current 
from  the  outside,  is  produced  in  the  rotor  by  e.m.f.  generated  therein. 
Moreover,  the  current  in  the  rotor,  instead  of  being  direct,  is  a  low-fre- 
quency alternating  current. 

Just  how  this  current  is  produced  in  the  rotor  will  be  apparent  from 
Fig.  7.  This  is  intended  to  represent  a  portion  of  a  squirrel-cage  rotor 
winding.  The  observer'is  supposed  to  be  looking  out  from  the  center 
of  the  rotor.  The  rectangles  marked  N  and  S  represent  the  poles  of  the 
stator,  passing  from  right  to  left  over  the  stator  and  rotor.  These  may 
be  thought  of  as  the  actual  poles  of  a  field  magnet.  As  a  matter  of  fact, 


ELEMENTARY  THEORY  7 

in  the  induction  motor  the  poles  do  not  exist  in  a  physical  sense,  but  the 
passing  of  the  magnetism  produces  substantially  the  same  effect  as 
would  the  passage  of  actual  poles.  The  principal  difference  is  that  in 
the  induction  motor  the  limits  of  the  poles  are  not  definite  as  shown,  but 
the  magnetism  decreases  gradually  from  a  maximum  to  zero,  and  then 
increases  in  the  opposite  direction. 

It  will  be  seen  from  Fig.  7  that  an  alternating  e.m.f .  will  be  generated 
in  the  conductors  of  both  the  rotor  and  the  stator  by  the  moving  flux. 
This  e.m.f.  will  be  strongest  in  those  conductors  at  the  middle  of  the 
poles,  and  will  become  less  as  they  move  from  the  middle,  zero  at  the 
point  midway  between  the  poles,  and  a  maximum  in  the  opposite  direc- 
tion at  the  middle  of  the  next  pole.  The  magnitude  of  this  e.m.f.  is  pro- 
portional to  the  flux  cut,  and  to  the  relative  speed  of  the  conductor  and 
the  flux.  It  will  therefore  be  greatest  when  the  rotor  is  at  rest,  will 
decrease  as  the  rotor  speed  increases,  and  will  become  zero  at  syn- 
chronism, i.e.,  when  the  rotor  is  moving  at  the  same  rate  as  the  flux. 

The  frequency  of  the  rotor  e.m.f.  is  also  variable.  At  rest  it  is  evi- 
dently the  same  as  the  stator  frequency.  As  the  speed  of  the  rotor 
increases,  the  cutting  will  be  less  rapid,  since  the  rotor  is  now  moving 
in  the  same  direction  as  the  magnetism.  At  the  synchronous  speed,  the 
cutting  is  zero  and  consequently  the  frequency  is  zero  also. 

The  currents  set  up  by  these  e.m.f  .'s  must  next  be  considered.  If  the 
rotor  had  only  resistance  but  no  reactance,  the  current  and  the  e.m.f. 
would  be  in  the  same  phase,  and  might  be  represented  by  the  same 
arrows.  Since,  however,  the  rotor  has  reactance,  the  current  will  lag 
behind  the  e.m.f. 

It  will  be  seen  at  once  that  the  currents  in  the  rotor  are  in  such  a 
direction  as  to  produce  a  torque  in  connection  with  the  stator  magnetism. 
Consequently,  the  rotor  will  start  from  rest.  As  the  speed  increases, 
however,  the  e.m.f.  generated  in  the  rotor  will  decrease,  as  was  just 
pointed  out,  and  will  become  zero  at  synchronism.  Hence,  the  rotor 
will  not  quite  reach  this  speed,  but  will  attain  only  such  a  speed  that  the 
current  will  be  just  sufficient  to  develop  the  torque  needed  to  maintain 
the  rotation.  If  there  is  no  load  on  the  motor  this  speed  will  be  very 
near  synchronism,  differing  from  it  by  only  a  small  fraction  of  one  per 
cent.  When  the  load  is  heavy,  however,  the  difference  between  the 
speed  and  the  synchronous  speed  must  be  greater,  and  will  amount  to 
from  2  to  10  per  cent  in  the  majority  of  cases.  This  difference  between 
the  synchronous  speed  and  the  actual  speed,  divided  by  the  former,  is 
called  the  slip. 


8 


THE  INDUCTION  MOTOR 


In  the  above  explanation  the  rotor  was  assumed  to  be  of  the  squirrel- 
cage  type.  It  might,  instead,  have  been  assumed  as  of  the  wound  rotor 
type.  In  this  case  it  would  have  been  provided  with  a  three-phase 
winding  similar  to  that  upon  the  stator.  The  e.m.f.'s  generated  would 
have  been  the  same,  but  the  currents  would  have  been  somewhat  different. 
In  Fig.  7  the  current  is  practically  free  to  take  any  path  it  chooses.  In 
the  case  of  the  wound-rotor  machine,  it  is  constrained  to  follow  through 
the  conductors  in  the  order  in  which  they  are  connected.  Thus,  in  the 
squirrel-cage  rotor  the  current  differs  from  bar  to  bar.  In  the  wound 
rotor  there  are  a  certain  number  of  definite  bands  of  current,  the  current 
being  constant  all  over  each  of  these  bands.  This  fact  is  in  itself  some- 
what disadvantageous,  but  the  use  of  this  type  of  winding  presents  cer- 
tain advantages,  as  will  be  pointed  out  presently. 


SHAPE  OF  FLUX  WAVE 

In  the  foregoing,  it  was  assumed  that  the  stator  current  was  the  only 
current  acting.  This  would  be  nearly  the  case  if  the  rotor  had  no  wind- 
ing, or  if  it  were  of  the  wound-rotor  type,  and  the  rotor  circuit  were  open. 
In  the  case  of  a  motor  in  actual  operation,  neither  of  these  suppositions 

is  true.  We  must  now  consider 
the  effect  of  the  rotor  currents  in 
modifying  the  distribution  of 
the  flux  which  would  be  given 
by  the  stator  current  alone. 

If  we  pass  three  direct  currents 
of  the  value  of  the  components  of 
the  three  phases  shown  in  Fig. 
4,  at  the  time  i,  the  magneto- 
motive forces  acting  across  the 
gap  will  be  represented  by  the 
broken  line  of  Fig.  8.  The  curve 
of  flux  would  be  of  the  same 
general  shape,  but  would  have 

the  corners  somewhat  rounded  on  account  of  leakage  of  the  flux.  On 
passing  to  the  current  at  the  point  2,  the  curve  changes  to  the  shape 
shown  in  Fig.  9,  and  becomes  flatter  on  top,  since  there  is  no  current  in 
the  phase  B  at  this  time.  At  the  point  3  the  shape  is  again  that  of  Fig.  8, 
and  so  on.  These  considerations  would  then  tend  to  show  that  the  flux 
wave  is  continually  changing  from  the  shape  of  Fig.  8  to  that  of  Fig.  9. 


FIG.  8. — Magneto  Motive  Force  Curve  of 
Three-phase  Induction  Motor. 


ELEMENTARY  THEORY 


9 


FIG.  9. — Magneto  Motive  Force  Curves  of 
Three-phase  Induction  Motor. 


It  will  be  noted  that  since  a  complete  cycle  of  the  change  takes  place  in 
60  degrees,  its  frequency  is  six  times  that  of  the  frequency  of  the  circuit. 
As  a  matter  of  fact,  however,  numerous  tests  show  that  when  the  applied 
wave  of  e.m.f.  is  a  sine  wave,  the 
curve  of  flux  is  also  of  the  sine 
shape  and  revolves  with  a 
uniform  velocity.  There  are 
several  reasons  for  this  fact. 

In  the  first  place,  the  rate  of 
cutting  of  the  flux  is  determined 
by  the  shape  of  the  wave  of 
applied  e.m.f.  and  not  directly 
by  the  shape  of  the  current  wave. 
This  is  true  on  account  of 
the  fact  that  in  any  motor  or 
transformer  the  counter  e.m.f.  is  almost  equal  to  the  applied  e.m.f.  The 
difference  is  only  enough  to  maintain  the  current,  and  this  difference  is 
always  small  compared  to  the  applied  e.m.f.  Consequently  if  the  applied 
e.m.f.  is  a  sine  wave,  the  counter  e.m.f.  is  also  practically  of  the  sine  shape. 

A  sine  wave  of  rotating  flux,  revolving  with  uniform  velocity,  will 
generate  a  sine  wave  of  e.m.f.  in  a  single  conductor.  Consequently,  it 
will  also  generate  a  sine  wave  of  e.m.f.  in  any  combination  of  conductors, 
since  any  number  of  sine  waves  of  the  same  frequency  added  together 
at  any  phase  relations  will  still  be  a  sine  wave.  A  sine  wave  of  flux  there- 
fore corresponds  to  the  most  general  case.  It  is  not  contended  that  the 
sine  wave  of  flux  is  the  only  form  of  wave  which  can  produce  a  sine  wave 
of  e.m.f.  provided  certain  groupings  of  conductors  were  adopted,  but  it 
is  evident  that  it  is  the  only  one  which  will  always  do  so.  Hence,  it  is 
the  one  which  should  be  used  in  general  calculations. 

It  may  not  be  evident  at  first  sight  how  a  sine  wave  of  flux  can  be 
generated  with  the  currents  as  shown  in  Fig.  5.  In  the  first  place,  the 
currents  will  not  follow  a  sine  shape.  This  is  especially  true  at  no  load. 
The  shape  of  the  current  wave  is  determined  by  the  difference  between  the 
curves  of  applied  and  counter  e.m.f.  The  difference  of  the  two  waves 
may  depart  quite  markedly  from  the  sine  shape,  without  either  of  the 
two  original  waves  being  materially  distorted,  and  consequently  the 
current  curve  may  be  distorted  without  serious  distortion  of  either  of 
the  two  e.m.f.  waves. 

The  influence  of  the  rotor  currents  on  the  stator  flux  is  also  of  great 
importance.  Imagine  the  rotor  operating  at  synchronism.  There  is 


10  THE  INDUCTION  MOTOR 

no  cutting  of  the  flux  by  the  rotor  conductors  and  consequently  no  cur- 
rent in  the  rotor  so  long  as  the  shape  of  the  flux  wave  does  not  change. 
It  is  evident,  however,  that  the  slightest  change  in  the  shape  of  the  flux 
curve,  or  the  slightest  change  hi  its  rate  of  rotation  will  at  once  cause  an 
e.m.f.  to  be  generated,  and  since  the  resistance  and  the  reactance  of  the 
rotor  are  both  low,  the  resulting  current  will  be  large.  According  to  the 
general  law  of  induction,  the  direction  of  the  secondary  current  will  be 
such  as  to  tend  to  stop  the  change  of  flux  producing  it.  This  effect,  it  will 
be  noted,  is  particularly  strong  in  the  case  of  low-resistance  squirrel-cage 
rotors,  and  less  so  in  wound  rotors  and  hi  cases  where  there  is  consider- 
able resistance  in  the  secondary  circuits.  A  rotor  having  many  bars  will 
be  more  efficient  hi  preventing  fluctations  than  one  having  only  a  few. 
This  is  true,  since  any  shif  ting  of  the  flux  within  a  tooth  of  the  rotor  may 
take  place  without  setting  up  a  corrective  current,  so  long  as  the  total 
flux  through  the  tooth  does  not  change.  Obviously,  the  smaller  the 
tooth,  the  less  the  possible  fluctuation. 

VECTOR  RELATIONS  OF  CURRENT  AND  E.M.F. 

We  are  now  in  a  position  to  return  to  Fig.  7,- and  consider  more  in 
detail  the  relations  of  the  current  and  the  e.m.f.  Let  us  assume  first 
that  the  rotor  is  operating  at  synchronism.  This  would  require  the  appli- 
cation of  a  small  amount  of  power  to  the  rotor  shaft,  to  overcome  the 
friction  and  windage.  This  condition  is  practically  realized  in  the  case 
of  a  motor  operating  without  load,  and  having  a  low  resistance  secondary. 
The  applied  e.m.f.  will  be  considered  to  be  of  the  sine  shape.  This  will 
always  be  assumed  to  be  the  case  unless  the  contrary  is  stated.  The 
distribution  of  the  flux  in  the  gap  will  therefore  be  sinusoidal,  as  just 
explained. 

Since  the  rotor  is  operating  at  synchronism,  there  is  no  cutting  of  the 
flux  by  the  rotor  conductors,  and  consequently  there  is  no  rotor  current 
except  the  small  currents  caused  by  the  attempt  of  the  flux  to  change  its 
shape.  These  will  have  only  a  small  effect,  and  will  be  neglected.  The 
e.m.f.  generated  in  the  stator  conductors  by  the  flux  may  be  represented 
by  the  arrows  on  the  rotor  bars.  So  long  as  the  same  flux  cuts  the  stator 
and  the  rotor  conductors,  the  e.m.f.'s  generated  in  the  two  will  be  strictly 
in  phase.  This  statement  refers  to  the  counter  generated  e.m.f.,  not  to 
the  applied  e.m.f.  It  is  evident  from  the  figure  that  the  generated  e.m.f. 
is  90  degrees,  or  one-quarter  period  behind  the  flux.  This  is  a  general 
rule,  and  is  always  true  hi  induction  motors,  alternators,  transformers, 


ELEMENTARY  THEORY 


11 


etc.  Thus  in  the  single-stator  coil  shown  in  Fig.  7,  the  e.m.f.  is  at  the 
instant  a  maximum  while  the  flux  is  zero.  It  is  true  that  flux  passes 
into  the  coil  from  both  poles  shown,  but  it  also  passes  out  again  before 
surrounding  either  conductor.  Hence,  none  passes  through  the  coil,  or 
we  say  the  flux  in  the  coil  is  zero.  The  generated  e.m.f.  therefore  lags 
behind  the  flux  by  90  degrees. 

In  speaking  of  the  angle  between  the  flux  and  the  e.m.f.,  we  make  the 
convention  that  a  flux  and  the  current  producing  it  are  in  the  same  phase 
when  they  reach  their  maxima  at  the  same  time.  This  is  sometimes 
expressed  as  equality  of  time-phase,  in  distinction  to  space-phase.  Thus, 
we  consider  a  flux  as  in  the  same  time-phase  as  the  current  producing 
it,  although  they  are  actually  at  right  angles  in  space. 

When  the  motor  is  operating  as  above  at  no  load  and  at  synchronous 
speed,  the  current  in  the  stator  is  determined  almost  entirely  by  the  mag- 
netomotive force  necessary  to  maintain  the  flux  across  the  gap.  This 
current  can  therefore  be  reduced  by  making  the  gap  as  short  as  possible. 
The  magnitude  of  this  current  will  in  general  be 
from  15  to  50  per  cent  of  the  full  load  current.  To 
keep  it  within  even  these  bounds  it  is  necessary  to 
use  very  short  gaps.  These  will  range  from  0.02  in. 
or  even  less,  to  0.125  in. 

In  Fig.  10  is  shown  the  vector  diagram  of  the 
motor  when  operating  at  synchronism  and  under 
no  load.  Since  the  only  current  acting  is  that  in 
the  stator,  the  flux  is  in  phase  with  the  current. 
This  is  a  general  rule  of  all  magnetic  circuits. 
Where  more  than  one  current  acts  on  a  given 
magnetic  circuit,  the  flux  will  be  in  phase  with  the 
resultant  of  all  the  currents  acting.  Thus  in  an 
induction  motor  under  load,  the  flux  will  be  in 
phase  with  and  proportional  to  the  resultant  of  both 
the  stator  and  the  rotor  magnetomotive  forces.  It 
is  true  that  there  is  a  slight  phase  difference  due  to 
hysteresis,  but  this  is  small  enough  to  be  neglected 
for  our  present  purposes.  The  student  should 
thoroughly  understand  these  two  principles,  that 
the  induced  e.m.f.  lags  90  degrees  behind  the  flux,  and  that  the  flux  is 
in  phase  with  the  resultant  of  all  the  currents  acting.  All  the  vector 
diagrams  are  based  on  these  two  simple  facts. 

In  Fig.  10  the  horizontal  line  is  taken  as  the  axis  of  the  flux,  and  since 


FIG.  10.  —  No-load 
Vector  Diagram  of 
Induction  Motor. 


12  THE  INDUCTION  MOTOR 

the  current  is  in  phase  with  the  flux,  it  is  the  current  axis  as  well.  The 
rotation  of  the  vectors  might  be  taken  in  either  direction.  For  our  pur- 
poses we  will  make  the  more  common  assumption  of  counter-clockwise 
rotation.  The  generated  e.m.f.  in  both  the  stator  and  rotor  will  be 
represented  by  a  vector  90  degrees  behind  the  flux,  or  by  OEi.  Since 
there  is  no  cutting  of  the  flux,  the  rotor  e.m.f.  is  zero,  but  the  line  OEi 
indicates  its  direction  under  all  circumstances  of  load.  The  primary 
applied  e.m.f.  will  be  represented  by  a  line  equal  and  opposite  to  OEi  or 
OE.  At  a  matter  of  fact,  the  applied  e.m.f.  will  of  course  have  to  be 
somewhat  larger  than  the  counter  e.m.f.,  and  at  such  an  angle  to  it  as  to 
maintain  the  required  current.  The  difference,  however,  is  not  large, 
and  for  the  present  it  will  be  taken  as  equal  and  opposite. 

There  is  another  small  inaccuracy  hi  the  diagram  as  drawn.  The 
power  supplied  as  shown  would  be  zero,  since  the  current  and  e.m.f.  are 
90  degrees  apart  in  phase,  and  the  power  is  given  by  El  cos  6.  This  is 
not  strictly  the  case,  since  a  small  amount  of  power  must  be  supplied  to 
make  up  for  the  losses  in  the  iron  and  copper. 

Now  consider  that  full  load  is  put  on  the  motor.  Under  these  cir- 
cumstances it  will  not  rotate  at  synchronous  speed,  but  will  drop  behind 
enough  so  that  sufficient  e.m.f.  will  be  generated  in  the  rotor  to  maintain 
the  current  necessary  to  produce  the  torque.  The  flux  will  remain  prac- 
tically the  same,  since  is  it  always  just  sufficient  to  generate  a  counter 
e.m.f.  almost  equal  to  the  applied  e.m.f.  The  difference  between  the 
two  must,  however,  be  greater  now  than  when  the  motor  was  unloaded, 
since  there  is  more  current.  The  flux  will  therefore  decrease  about  3 
per  cent  in  a  moderate-size  motor. 

The  e.m.f.'s  in  the  rotor  and  stator  will  have  the  same  relation  to  the 
flux  as  in  the  case  of  no  load.  The  principal  difference  is  that  we  now 
have  current  in  the  rotor.  The  phase  relation  of  this  current  to  the 
e.m.f.  of  the  rotor  is  of  the  greatest  importance.  In  a  circuit  carrying 
an  alternating  current,  the  current  will  lag  behind  the  applied  e.m.f. 
whenever  the  circuit  has  inductive  reactance  as  well  as  resistance.  A 
circuit  has  inductive  reactance  whenever  there  are  leakage  lines  of  mag- 
netic flux  cutting  the  circuit  and  consequently  generating  in  it  an  e.m.f.  in 
addition  to  the  applied  e.m.f.  This  generated  e.m.f.  is  always  in  such  a 
direction  as  to  tend  to  make  the  current  lag.  The  rotor  has  such  a 
leakage  flux,  since,  on  account  of  the  current  in  the  end  rings  and  in 
the  bars  where  they  project  beyond  the  slots,  flux  surrounds  these'  parts. 
There  is  also  some  rotor  leakage  flux  around  the  bars  within  the  slots. 
All  these  leakage  fluxes  cause  the  current  to  lag. 


ELEMENTARY  THEORY 


13 


It  will  be  remembered,  however,  that  the  frequency  of  the  current 
in  the  rotor  is  not  constant,  but  varies  from  zero  at  synchronism  to  the 
line  frequency  at  standstill.  The  magnitude  of  the  e.m.f.  generated, 
due  to  this  leakage  flux,  will  therefore  be  variable,  being  proportional  to 
the  rotor  frequency  and  consequently  to  the  slip.  The  lag  of  the  rotor 
current  behind  the  rotor  e.m.f.  will  likewise  be  variable,  being  zero  at 
synchronism  and  nearly  90  degrees  at  standstill. 

As  soon  as  we  put  load  on  a  motor,  current  is  produced  in  the  rotor, 
and  this  current  tends  to  produce  a  flux  through  the  air  gap.  This  flux 
would  be  in  the  same  direction  as  the  rotor  current  and  consequently 
nearly  at  right  angles  to  the  main  flux.  It  would  therefore  tend  in 
general  to  increase  the  total  flux.  We  have  seen,  however,  that  this  flux 
cannot  change  materially,  since  the  counter  e.m.f.  must  always  be 
approximately  equal  to  the  applied  e.m.f.  The  total  resultant  m.m.f.  of 
the  entire  magnetic  circuit  must  therefore  remain  nearly  constant,  and 
must  also  be  in  phase  with  the  flux.  What  happens,  therefore,  is  that 
the  flux  decreases  slightly,  just  enough  to  allow  sufficient  additional 
current  to  flow  in  the  stator  to  offset  the  magnetizing  action  of  the 
rotor  current.  For  simplicity  we  may  assume  that  the  number  of  con- 
ductors in  the  rotor  is  the  same  as  the 
number  in  the  stator.  In  this  case  the 
extra  current  in  the  stator  will  be  exactly 
opposite  to  that  in  the  rotor.  In  any 
event,  the  added  m.m.f.  of  the  stator 
current  will  be  equal  and  opposite  to  the 
rotor  m.m.f. 

As  we  have  shown,  the  primary 
applied  e.m.f.  is  approximately  90  degrees 
ahead  of  the  flux.  The  rotor  current  lags 
somewhat  more  than  90  degrees  behind 
the  flux.  Since  the  added  primary 
current  is  equal  and  opposite  to  the 
rotor  current,  it  must  lead  the  flux 
somewhat  less  than  90  degrees.  The 
total  stator  current  is  the  vector  sum  of 
the  magnetizing  current  and  the  com- 
ponent which  offsets  the  rotor  current. 

Fig.  ii  shows  the  relation  of  these  quantities.  As  is  always  the  case, 
the  applied  e.m.f.  is  approximately  90  degrees  ahead  of  the  flux.  The 
rotor  current  is  indicated  by  7r,  lagging  somewhat  behind  the  rotor  e.m.f. 


FIG.     ii.  —  Full-load    Vector 
Diagram  of  Induction  Motor. 


14 


THE  INDUCTION  MOTOR 


The  primary  current  is  represented  by  I8,  and  is  of  such  a  value  as 
to  offset  the  rotor  current,  and  at  the  same  time  supply  the  magnetizing 
current  Im. 

STARTING  CONDITIONS 

When  the  rotor  is  at  rest  and  power  is  applied  to  start  it,  the  frequency 
in  the  rotor  is  the  same  as  in  the  stator.  Hence  it  is  evident  that  the  lag 
of  the  current  behind  the  e.m.f.  will  be  great.  In  fact,  if  the  rotor  resist- 
ance were  zero,  the  lag  would  be  90  degrees.  With  an  ordinary  squirrel- 
cage  rotor  the  lag  will  be  about  70  degrees.  The  current  is  of  course 


FIG.  12. — Vector  Diagram  of  Induction  Motor  at  Moment  of  Starting. 

large,  and  since  the  stator  must  carry  a  correspondingly  large  component 
of  current  to  offset  this  rotor  current,  the  power-factor  of  the  motor  is 
very  low.  The  vector  diagram  for  this  case  is  shown  in  Fig.  12. 

The  torque  of  any  motor  is  proportional  to  the  product  of  the  flux 
and  the  rotor  current.  It  is  also  influenced  by  the  angle  between  the 
two,  being  a  maximum  when  the  angle  is  90  degrees,  and  zero  when  it 
is  zero  degrees.  This  is  readily  seen  in  the  case  of  a  shunt-wound, 
direct-current  motor.  If  the  armature  be  so  placed  that  the  current  passes 
first  to  those  conductors  situated  midway  between  the  poles,  i.e., 
if  the  brushes  are  in  the  neutral  position,  the  torque  will  be  a  maximum 
(this  neglects  the  distorting  action  of  the  armature  current).  If,  on  the 
other  hand,  the  brushes  are  placed  90  degrees  from  this  position,  the 
torque  will  be  zero.  The  same  thing  is  true  of  the  induction  motor. 


ELEMENTARY  THEORY  15 

Consequently  the  torque  will  be  greatest  for  a  given  rotor  current  and  a 
given  flux  when  the  rotor  current  is  90  degrees  behind  the  flux  or  in 
phase  with  the  e.m.f .  generated  in  the  rotor. 

From  the  foregoing,  it  will  be  seen  that  the  starting  torque  for  a  given 
current  will  be  small  in  a  motor  so  constructed  that  the  lag  of  the  rotor 
current  is  large.  This  is  the  case  with  the  ordinary  squirrel-cage  motor. 
It  is  true  that  every  effort  is  made  to  reduce  the  reactance  of  the  rotor, 
but  it  is  also  true  that  the  resistance  of  the  rotor  must  be  kept  low  in 
order  that  the  copper  loss  and  the  slip  may  be  low.  In  order  to  provide 
even  moderate  starting  torque,  it  is,  however,  necessary  to  make  the 
resistance  of  the  rotor  much  higher  than  would  otherwise  be  desirable. 
This  is  done  by  the  use  of  brass  end  rings  instead  of  copper,  by  using 
copper  rings  of  small  section,  or  by  other  suitable  means.  It  is  obvious 
that  the  best  performance  requires  large  rotor  resistance  during  starting 
and  low  running  resistance.  This  condition  is  frequently  secured  by  the 
use  of  a  rotor  provided  with  a  winding  similar  to  the  stator  winding,  the 
terminals  being  brought  to  three  slip  rings,  and  connected  to  external 
resistances  or  short-circuited  as  occasion  may  require.  This  constitutes 
what  is  known  as  a  wound-rotor  machine.  It  is  inferior  to  the  squirrel- 
cage  motor  in  every  respect,  except  that  it  has  improved  starting  prop- 
erties and  that  the  speed  is  readily  adjustable.  The  cost  is  of  course 
higher.  On  account  of  these  facts  probably  three  out  of  every  four 
motors  sold  are  of  the  squirrel-cage  type.  This  subject  of  starting 
torque  will  be  treated  more  fully  in  a  later  chapter. 


CHAPTER   II 
THEORY  OF  THE  INDUCTION  MOTOR 

IN  the  previous  chapter  we  have  considered  somewhat  the  elementary 
theory  of  the  induction  motor.  It  is  now  necessary  to  take  into  account 
the  effect  of  the  magnetic  leakage  in  the  stator  and  the  rotor.  There  are 
three  fluxes  that  we  must  consider.  These  are  the  flux  passing  into 
both  stator  and  rotor,  the  flux  in  the  stator  alone  and  the  flux  in  the  rotor 
alone.  Of  these  three  the  first  is  the  useful  flux  of  the  motor.  The 
other  two  are  the  leakage  fluxes  and  should  be  reduced  to  as  small  pro- 
portions as  possible.  The  performance  of  the  motor  can  perhaps  best 
be  understood  by  considering  it  as  a  transformer.  It  can  readily  be 
shown  that  if  in  the  case  of  a  wound-rotor  machine,  the  rotor  be  held 
stationary  and  the  resistance  in  the  rotor  circuit  be  varied,  the  perform- 
ance of  the  machine  in  its  general  electrical  respects  is  the  same  as 
though  the  rotor  rings  were  short-circuited,  the  machine  were  in  motion, 
and  the  load  on  the  motor  were  varied.  This  fact  enables  us  to  treat 
the  induction  motor  in  all  respects  as  though  it  were  a  transformer,  and 
leads  to  a  considerable  simplification  of  the  calculations.  It  also  has 
the  obvious  advantage  of  developing  the  theory  of  the  transformer  with- 
out extra  labor. 

Consider  a  transformer  supplying  current  to  a  non-inductive  circuit, 
or  an  induction  motor  with  its  rotor  at  rest  and  its  slip  rings  connected 
to  non-inductive  resistors.  The  resistors  should  be  such  as  to  give  a  bal- 
anced secondary  load.  Let  the  e.m.f .  per  rotor  circuit  be  E,  the  rotor 
resistance  Rr,  the  external  resistance  R,  and  the  rotor  inductance  Lr.  If 
then  the  frequency  be  /,  the  angular  velocity  of  the  rotating  field  will  be 
o)=27r/.  We  then  have  for  the  rotor  current  per  phase, 


/= 


In  the  case  of  the  motor  with  the  rotor  revolving  in  the  usual  way,  we 
have  a  similar  equation,  but  we  have  to  take  account  of  the  fact  that  both 
the  rotor  e.m.f.  and  the  rotor  frequency  are  less  than  in  the  case  of  the 

16 


THEORY  OF  THE  INDUCTION  MOTOP  17 

motor  with  stationary  rotor.  This  is  due  to  the  fact  that  both  the  rota- 
ting magnetic  field  and  the  rotor  are  revolving  in  the  same  direction,  and 
hence  the  rate  of  cutting  is  less.  For  example,  if  the  rotor  is  running  at 
95  per  cent  of  its  synchronous  speed,  the  rate  of  cutting  will  obviously 
be  only  5  per  cent  as  great  as  would  be  the  case  if  the  rotor  were  at  rest. 
Likewise  the  frequency  will  be  only  5  per  cent  of  the  primary  frequency. 
The  equation  for  the  rotor  current  is  then 


where  s  is  the  percentage  of  slip.    This  can  readily  be  changed  to 
the  form, 

E 


1= 


The  expressions  in  the  case  of  the  induction  motor  and  in  that  of  the 
transformer  are  the  same,  except  for  the  term  denoting  the  resistance. 
In  the  case  of  the  transformer  this  term  is  (Rr+R),  and  in  the  case  of  the 

D 

motor  it  is  — -.     Hence  it  is  evident  that  if  instead  of  allowing  a  motor 

to  rotate  freely  with  a  slip  s  and  a  rotor  resistance  Rr,  we  hold  the  rotor 
so  it  cannot  revolve,  and  add  enough  external  resistance  so  the  total 

D 

resistance  is  equal  to  — ,  all  the  electrical  conditions  will  be  the  same  as 

before,  and  we  can  treat  the  motor  in  all  respects  as  a  stationary  trans- 
former. 

Another  great  simplification  of  our  equations  can  be  made  by  assum- 
ing a  suitable  number  of  turns  on  the  rotor.  For  example,  it  will  be  evi- 
dent at  once  that  doubling  the  number  of  turns  on  the  rotor  and  at  the 
same  time  making  the  cross-section  of  the  conductor  half  as  great,  would 
in  no  way  change  the  electrical  characteristics  of  the  machine.  Hence, 
in  general  we  are  at  liberty,  in  developing  the  theory,  to  choose  the  ratio 
of  the  rotor  to  stator  winding  which  gives  the  simplest  relation.  For 
this  reason  we  choose  the  ratio  of  one  to  one. 

An  ideal,  perfect  transformer,  i.e.,  one  with  no  losses  and  with  no 
magnetic  leakage  in  either  primary  or  secondary,  has  no  effect  on  the 
circuit  in  which  it  is  inserted,  provided  the  ratio  is  one  to  one.  The  cur- 
rent and  the  e.m.f .  in  the  secondary  will  be  exactly  equal  to  the  primary 
current  and  e.m.f.,  and  there  will  be  no  phase  displacement  between 


18  THE  INDUCTION  MOTOR 

the  primary  and  secondary  currents,  since  there  will  be  no  leakage  flux, 
i.e.,  flux  surrounding  only  one  of  the  windings.  This  is  also  evident 
from  the  fact  that  since  there  are  no  losses,  and  since  the  currents  and 
the  e.m.fs.  are  equal,  the  power-factors  on  the  two  sides  must  be  equal. 
Such  a  transformer  could  then  be  replaced  by  a  wire  of  zero  resistance 
and  inductance. 

If  now  we  take  sucn  an  ideal  transformer  and  add  to  it  the  elements 
we  have  neglected,  we  shall  have  an  apparatus  which  will  act  in  all 
respects  like  the  ordinary  commercial  transformer.  The  local  resistance 
and  reactance  of  the  primary  and  secondary  can  be  replaced  by  coils 
having  the  same  reactance  and  resistance.  The  core  loss  we  may  replace 
by  a  non-inductive  resistance  consuming  the  same  amount  of  power  at 
the  same  voltage  as  the  core  loss,  and  we  can  use  an  inductance  having 
no  resistance  to  take  a  current  equal  to  the  magnetizing  current.  The 
equivalent  circuit  of  such  a  transformer  is  shown  in  Fig.  13. 


FIG.  13. — Equivalent  Electric  Circuits  of  Transformer  or  Induction  Motor. 

This  diagram  gives  an  exact  equivalent  of  the  circuit  of  a  commer- 
cial transformer,  with  the  following  slight  exceptions.  It  is  not  exact  if 
the  voltage  varies,  since  in  the  equivalent  circuit  the  magnetizing  current 
passing  through  Lc  is  proportional  to  the  voltage.  In  the  commercial 
transformer  this  is  not  strictly  true  on  account  of  saturation  of  the  mag- 
netic circuit.  Likewise  in  the  equivalent  circuit,  the  loss  corresponding 
to  the  core  loss,  i.e.,  the  loss  in  Rc,  is  proportional  to  the  square  of  the 
voltage.  This  is  true  of  that  part  of  the  core  loss  due  to  eddy  currents, 
but  is  not  true  of  that  part  due  to  hysteresis.  However,  the  discrepancy 
is  slight,  and,  moreover,  both  transformers  and  induction  motors  are 
usually  operated  at  constant  potential.  Their  deviations  from  the  exact 
facts,  therefore,  introduce  no  appreciable  error. 

The  only  important  divergence  of  the  theory  from  the  facts  is  that 
in  developing  the  equations  we  treat  the  inductance  of  the  primary  and 
secondary  as  though  they  were  constant.  This  is  not  entirely  true  in 
the  actual  transformer,  since  on  account  of  magnetic  saturation,  the 


THEORY  OF  THE  INDUCTION  MOTOR  19 

inductance  becomes  less  as  the  current  increases.  However,  a  large 
part  of  the  path  of  the  leakage  lines  is  through  the  air,  and  the  change  in 
inductance  is  slight  and  may  well  be  neglected. 

It  is  entirely  possible  to  treat  the  circuit  of  Fig.  13  analytically,  and 
the  conclusions  will  be  exact  except  in  the  slight  particulars  noted;  but  a 
further  modification  of  the  diagram  is  possible  which,  while  not  intro- 
ducing any  serious  error,  will  greatly  simplify  our  calculations.  This 
change  consists  in  considering  that  the  magnetizing  current  and  the  core- 
loss  current  are  taken  directly  from  the  line,  instead  of  having  to  pass 
first  through  the  resistance  and  inductance  of  the  primary.  The 
modified  circuit  is  shown  in  Fig.  14.  Since  the  magnetizing 
current  and  the  core-loss  current  are  small,  the  error  introduced  will  be 
negligible. 

We  now  proceed  to  prove  that  with  this  arrangement  of  the  circuit, 
the  ends  of  the  current  vectors  of  both  the  primary  and  the  secondary 


FIG.  14. — Modified  Equivalent  Circuit  of  a  Transformer  or  Induction  Motor. 

currents,  if  plotted  with  their  proper  magnitude  and  phase  position,  will 
all  fall  on  a  circle.  This  generalization  is  of  the  greatest  value  to  the  stu- 
dent of  the  induction  motor  and  transformer,  as  it  not  only  enables  him 
to  scale  all  the  values  of  the  current,  etc.,  directly  from  a  simple  diagram, 
but  it  also  is  of  the  greatest  use  in  giving  a  remarkably  clear  mental  pic- 
ture of  the  relation  of  the  various  quantities  involved.  It  is  also  of 
value  in  the  testing  of  induction  motors. 

To  prove  this  fact  is  comparatively  simple.  Using  the  same  values 
as  in  Fig.  14,  the  current  in  the  part  of  the  circuit  beyond  Lc  and  Rc  is 
given  by  the  expression, 


7= 


This  is  the  current  in  the  secondary  (rotor).  The  current  in  the  primary 
(stator)  is  the  same  plus  the  currents  taken  by  Lc  and  Rc.  This  current 
must  of  course  be  added  vectorially.  The  secondary  current,  on  account 


20  THE  INDUCTION  MOTOR 

of  the  inductance  in  the  circuit,  lags  behind  the  e.m.f.     The  sine  of  the 
angle  of  lag  is  equal  to  the  reactance  divided  by  the  impedance,  or 


Substituting  this  value  in  the  expression  for  the  current,  we  get 
Esinfl 


7= 


a>Lp+Ls 

This  is  the  polar  equation  of  a  circle  of  diameter  —=  =.    That 
this  is  so  will  readily  appear  from  Fig.  15.     Taking  O  as  the  origin,  let 


0  A 

FIG.   15. — Vector    Relations  of  Circuit  with  Constant  Reactance  and  Variable 
Resistance. 

the  vertical  line  OE  represent  the  applied  e.m.f.  The  current  will  be 
represented  by  some  such  line  as  OI  behind  the  e.m.f.  by  the  angle  6. 
Through  the  points  O  and  /  draw  a  circle  tangent  to  the  line  OE,  and 
draw  the  line  I  A.  The  angle  I A  O  is  evidently  equal  to  6.  If  we 

assume  the  diameter  of  the  circle  OA  to  be  equal  to  ,it  is  evident 

uLp+Ls 

that  the  line  OI  will  be  equal  to  — — = —  — .     Hence,  the  expression  is 
w(LP+Ls) 

the  equation  of  a  circle  whose  diameter  is 


a>Lp-\-L8 

To  obtain  the  primary  current  we  must  add  to  the  current  just  found, 
the  current  through  L  and  R.    This  current  will  lag  behind  the  primary 


THEORY  OF  THE  INDUCTION  MOTOR 


21 


e.m.f.  by  such  an  angle  that  cos  0-- 


In  the  actual  motor 


this  angle  will  be  large,  since  in  most  cases  the  magnetizing  current 
lagging  90  degrees  will  be  considerably  larger  than  the  power  component 
of  the  current  to  supply  the  losses.  In  Fig.  16,  the  line  OB  is  drawn  to 
represent  this  current.  The  total  primary  current  then  is  the  vector  sum 
of  BO  and  OI  or  BI.  To  complicate  the  diagram  less,  the  vector  repre- 
senting the  e.m.f.  is  drawn  from  the  point  B. 

Several  things  are  at  once  apparent  from  the  diagram.  If  we  take 
any  value  of  the  primary  current  as  BI,  we  can  separate  it  into  two  com- 
ponents ID  and  7C,  the  first  parallel  to  the  e.m.f.  and  the  second  perpen- 
dicular to  it.  The  former  is  the  power  component  of  the  primary  cur- 
rent, the  latter  is  the  wattless  component.  By  using  the  proper  scale, 


FIG.  16.— Circle  Diagram  of  the  Induction  Motor. 

it  is  evident  that  the  power  component  of  the  current  can  be  considered 
as  the  power  itself.  This  is  true  only  when,  as  is  usually  the  case,  the 
e.m.f.  is  constant.  Thus  at  the  current  7,  the  power  input  is  BC,  and  of 
this,  the  portion  BG  is  wasted  in  the  core  loss.  The  maximum  power 
input  into  the  primary  is  evidently  attained  when  the  current  is  repre- 
sented by  BJ.  The  power-factor  of  the  secondary  is  then  70  per  cent, 
or  the  angle  of  lag  of  the  secondary  current  is  45  degrees.  It  will  readily 
be  seen  that  this  is  equivalent  to  saying  that  the  reactance  and  resistance 
of  the  secondary  must  be  equal  for  maximum  input.  The  maximum 
value  of  the  power-factor  is  obtained  at  such  a  value  of  the  current  that 
the  current  vector  is  tangent  to  the  circle. 

We  have  previously  shown  that  the  construction  holds  equally  well 
in  the  case  of  the  induction  motor.  Its  application  to  the  testing  of  an 
actual  motor  is  shown  in  Fig.  17.  The  motor  had  a  rotor  of  the 
squirrel-cage  variety. 


22 


THE  INDUCTION  MOTOR 


Volts. 


440 


Amp. 
per  phase. 


TABLE  I 

NO-LOAD  READINGS 

Watts.         Power  factor  Power 

W  -T-  (\/~^EI  component 

of  current. 
3805  0.476  5.00 

LOCKED-ROTOR  READINGS 

Corrected  values 


componer 
of  curren 


9-23 


Volts.          Amp.       Watts.       Volts.         Amp. 


220  41-9         429O          44O  83.8        17160 

Res.  per  stator  phase  (Y  wound)  =0.397  ohm. 
Stator  loss  at  start=3Xo. 397X83. 84=8400  watts. 
Rotor  loss  at  513^=17160  —  8400—3805=4955  watts. 


Power      Power        Wattless 
factor  component  component 
of  current     of  current 
0.269       22.5  80.8 


IT       H  C  P 

FIG.  17. — Circle  Diagram  of  the  Induction  Motor. 

The  data  used  in  constructing  Fig.  17  are  given  in  Table  i.  In 
taking  these  data,  only  an  ammeter,  voltmeter,  and  wattmeter  were 
used.  Only  two  readings  were  taken,  one  with  the  rotor  running  freely 
without  load,  and  the  other  with  the  rotor  locked,  so  it  could  not  rotate. 
In  taking  this  latter  reading,  it  is  usually  impracticable  to  apply  the  full 
voltage  to  the  motor,  since  it  would  heat  up  very  rapidly  under  these  cir- 
cumstances. The  readings  were  obtained  by  applying  half  of  normal 
voltage  and  multiplying  the  current  observed  by  the  ratio  of  the  rated 
voltage  to  the  observed  voltage,  and  the  computed  watts  by  multiplying 
the  observed  watts  by  the  square  of  the  same  ratio.  The  power-factor 
is  obtained  in  the  usual  way  by  dividing  the  real  power  by  the  apparent 
power;  or, 

cos  0=-£_ 


THEORY  OF  THE  INDUCTION  MOTOR  23 

Taking  then  the  watts  consumed  at  no  load,  the  value  of  the  power  com- 
ponent of  the  current  is  given  by  the  expression  7  cos  0,  and  the  value  of 
the  wattless  component  is  similarly  given  by  /  sin  0.  These  two  com- 
ponents are  plotted  respectively  as  OH  and  OL.  Similarly,  the  two 
components  of  the  starting  current  are  determined  and  plotted  as  PR 
and  a  horizontal  line  (not  drawn)  from  R  to  EB.  This  line  is  equal  to 
BP.  Since  the  flux  is  approximately  constant  the  core  loss  will  also  be 
constant,  and  will  be  equal  to  the  distance  between  the  lines  LA  and  BP. 

Since  at  the  point  R  the  rotor  is  stationary,  the  output  is  zero  and  all 
the  power  put  into  the  motor  is  wasted.  The  line  PR  represents  the 
power  component  of  the  current,  or  to  a  proper  scale  the  power  itself. 
Hence  it  represents  the  total  loss  in  the  motor  at  standstill.  Of  this  loss 
the  portion  PS  is  the  stator  iron  loss.  The  remainder  must  be  copper 
loss  in  the  stator  and  rotor  and  the  rotor  iron  loss.  This  copper  loss  con- 
sists of  two  parts,  the  loss  in  the  stator  and  that  in  the  rotor.  Let  us 
therefore  divide  it  into  two  parts  by  a  point  T.  RT  is  then  the  rotor 
copper  loss  and  ST  is  the  stator  copper  loss,  or  more  exactly  the  copper 
loss  in  the  stator  due  to  the  component  of  the  stator  current  which  offsets 
the  rotor  current.  The  copper  loss  due  to  the  magnetizing  current  is 
already  included  in  the  loss  PS. 

If  we  join  the  points  R  and  T  to  the  point  O  by  straight  lines  we  can 
readily  determine  the  losses  at  any  load.  Thus  let  the  primary  current 
be  BI,  and  draw  the  vertical  line  1C.  Then  CG  is  the  core  loss  and  the 
copper  loss  due  to  the  no-load  current  BO,  GU  is  the  stator  loss  and  UK 
is  the  rotor  copper  loss.  The  total  input  into  the  machine  is  1C.  Tak- 
ing out  the  losses  just  mentioned,  the  remainder  IK  is  the  output  of  the 
machine.  The  efficiency  is  obviously  IK  divided  by  1C. 

That  the  total  copper  loss  with  the  exception  of  that  due  to  the 
component  representing  the  no-load  current  is  represented  by  the  inter- 
cept GK  may  be  readily  shown  as  follows.  What  we  have  to  prove  is 
that  the  intercept  SR  and  GK  have  the  same  ratio  as  the  squares  of  the 
lines  representing  the  respective  currents,  or, 

GK_  J 

SR~\OR 

But 

01  °L 

GK_OG=  OI  cos  (API)  'OA 

SR~~OS~ORcos(AOR)~  _p  OR 

R'OA 

therefore  the  proposition  is  proved. 


-(—}* 

(OR)' 


24  THE  INDUCTION  MOTOR 

A  number  of  other  quantities  can  be  at  once  taken  from  the  diagram. 
The  primary  power-factor  is  1C  divided  by  IB,  or  it  is  cos  6.  The 
maximum  power  output  of  the  motor  is  WV,  in  which  W  is  denned  by 
the  point  of  tangency  of  a  line  drawn  parallel  to  OR.  This  is  so,  since 
this  is  the  longest  possible  line  for  the  output.  The  maximum  input  is 
at  the  point  /.  (See  Fig.  16.)  It  might  seem  that  the  points  of 
maximum  input  and  output  should  be  the  same.  They  are  nearly  the 
same,  but  differ  somewhat  on  account  of  the  change  in  efficiency,  with 
the  load. 

SLIP  AND  TORQUE 

Before  showing  Kow  we  can  obtain  the  values  for  the  slip  and  the 
torque  from  the  diagram,  it  is  necessary  to  prove  two  propositions.  The 
first  of  these  is  as  follows:  The  slip  expressed  in  percentage  of  the  full- 
load  speed  is  equal  to  the  rotor  copper  loss  divided  by  the  input  to  the 
rotor.  As  has  been  previously  shown,  the  rotor  current  is  given  by 


The  rotor  copper  loss  is 


The  totai  input  into  the  rotor  is 

EsE  R  sE2R 

P  =  EI  cos  6=  — ^ 22  =  -fl2.    2/2  F 

The  percentage  of  rotor  loss  is  evidently  the  ratio  of  these  two,  or  S. 
Hence  the  proposition  is  proved. 

Torque  is  frequently  defined  in  synchronous  watts.  By  this  we 
mean  the  watts  that  would  be  developed  at  the  shaft  if  the  motor  were 
running  at  synchronous  speed,  and  exerting  the  given  torque.  It  is 
obvious  that,  if  more  convenient,  we  may  express  torque  in  synchronous 
kilowatts  or  in  synchronous  horse  power.  This  latter  is  a  very  conveni- 
ent way  of  expressing  the  torque.  Thus  if  we  say  a  certain  25-h.p. 
motor  develops  a  starting  torque  of  50  synchonous  h.p.,  we  know  at  once, 
without  calculation,  the  starting  ability  of  the  motor ;  that  is,  the  motor 
will  start  a  load  requiring  approximately  double  full-load  torque.  More- 
over the  torque  is  expressed  without  reference  to  the  speed. 

The  mechanical  output  of  the  motor  is  evidently  D  (i  —  S)  where  D  is 
the  synchronous  torque.  Since  the  percentage  of  rotor  loss  is  as  we  have 


THEORY  OF  THE  INDUCTION  MOTOR  25 

just  shown,  equal  to  S,  the  slip,  the  output  is  also  given  by  Input  X  (i  —  S) 
Writing  these  two  expressions  equal  to  one  another,  we  see  at  once  that 
Input  =  D.  Hence  we  may  state  the  general  rule,  the  synchronous 

o  p2  P 

torque  is  equal  to  the  total  rotor  input;  or  D=    2       2       „.     This  is 

true  whether  the  rotor  is  at  rest  or  in  motion.  It  must  be  kept  in  mind, 
however,  that  both  of  these  expressions  have  been  derived  on  the  supposi- 
tion that  we  have  sine  waves  of  current  and  flux.  If  these  conditions  are 
not  fulfilled,  the  torque  will  be  less  than  indicated  by  the  above.  In  any 
practical  case  we  have  also  to  make  some  deduction  for  the  torque 
required  to  overcome  friction.  This  is  in  general  a  small  correction. 
On  the  other  hand,  the  starting  torque  is  increased  very  materially  in 
many  cases  by  the  iron  losses  in  the  rotor.  This  is  explained  more  in 
detail  elsewhere.  In  consequence  of  these  facts,  the  writer  does  not 
consider  the  method  of  determining  the  starting  torque  by  computing 
the  rotor  lost  a  very  practical  one,  but  would  prefer  whenever  possible 
to  measure  the  torque  directly. 

From  the  two  above  propositions,  we  see  at  once  from  the  diagram 
that  the  percentage  of  slip  is  given  by  UK  divided  by  IU,  and  the  torque 
is  given  by  IU.  In  particular  the  starting  torque  is  equal  to  TR.  A 
motor  designed  to  give  the  greatest  possible  starting  torque  should  take 
a  starting  current  just  a  trifle  less  than  BJ,  Fig.  16. 

For  convenience  of  the  reader  we  add  a  table  of  the  various  quantities 
that  can  be  scaled  from  the  diagram  of  Fig.  17. 

BI=  primary  current; 
OI=  secondary  current; 
CI=  input  into  motor; 

CG=  fixed  loss,  i.e.,  hysteresis,  eddy  currents,  and  friction; 
GU=  primary  copper  loss; 
KU=  secondary  copper  loss; 
KI=  output  of  motor; 
KI  +  IC=  efficiency  of  motor; 
CI  -±-BI=  power  factor; 

BO=  idle  current; 
KU  +  UI=  slip; 

UI=  synchronous  torque; 
TR=  starting  torque  in  synchronous  watts; 
WV=  maximum  output  of  motor. 

Referring  to  the  simplified  circle  diagram  of  Fig.  18,  which  is  obtained 


26 


THE  INDUCTION  MOTOR 


from  the  form  just  given  by  assuming  that  the  no  load  losses  of  the  motor 
may  be  neglected,  the  ratio  of  the  magnetizing  current  to  the  rotor  current 
of  the  motor  with  the  rotor  locked  and  assuming  that  the  rotor  has  only 
reactance  and  no  resistance,  is  of  great  importance.  This  value  is  called 
the  leakage  coefficient,  and  is  usually  designated  by  the  letter  o,  or 

OA 
a=-~--.    The  determination  of  this  factor  is  treated  in  Chapter  (8). 

We  can  readily  derive  a  relation  between  this  factor  and  the  maximum 
value  of  the  power-factor  of  the  motor.  At  the  condition  of  maximum 
power-factor,  the  current  vector  is  represented  by  a  line  such  as  OC 
drawn  from  O  and  tangent  to  the  circle.  The  angle  of  lag  0  is  the  angle 


r 


FIG. 


EOC,  and  this  angle  is  equal  to  the  angle  CDO.     We  then  have 


AB 


AB 


and  since 


OD     20D     AB+20A     ^     2OA' 
OA 


cosO  = 


1  +  20 


Fig.  19  will  serve  to  illustrate  the  difficulties  encountered  in  attempt- 
ing to  construct  a  more  exact  diagram  based  on  the  connection  shown  in 
Fig.  13.  We  will  start  with  the  vector  E8  representing  the  total  voltage 
generated  in  the  rotor  of  an  induction  motor,  or  the  secondary  of  a  trans- 
former. Of  this,  a  portion  XgI8  is  used  up  in  overcoming  the  reactance 
of  the  secondary  circuit,  and  a  portion  R8I8  in  overcoming  the  resistance. 
In  the  transformer,  Xs  may  be  entirely  due  to  the  windings,  or  in  the 
more  general  case,  it  may  be  partially  due  to  the  windings  and  partially 


THEORY  OF  THE  INDUCTION  MOTOR 


27 


due  to  the  reactance  of  the  connected  load.  It  may  even  readily  happen 
that  Xg  is  negative,  if  the  transformer  is  supplying  over-excited  sychron- 
ous  machinery,  or  a  long  transmission  line,  taking  a  leading  current.  In 
the  induction  motor,  however,  Xs  is  in  general  entirely  due  to  the  react- 
ance of  the  windings  alone.  No  object  would  be  gained,  and  the  charac- 
teristics of  the  motor  would  be  seriously  impaired  by  the  use  of  additional 
reactance  in  this  circuit.  On  the  other  hand,  the  use  of  negative  reac- 
tance consisting  of  condensers  would  result  in  an  improvement  of  the 


FIG.  19. — Exact  Construction  for  Relation  of  Currents  and  e.m.fs.  of  an  Induction 
Motor. 


characteristics  of  the  motor.  It  is,  however,  impractical  to  make  use  of 
this  fact,  since  the  condensers  required  at  the  low  voltage  and  frequency 
of  the  rotor  would  be  of  prohibitive  size. 

Knowing  then  the  value  of  the  rotor  reactance,  and  the  current  in  the 
secondary,  we  can  at  once  determine  the  length  of  the  line  XSIS.  Its 
direction  will  be  determined  by  the  fact  that  its  end  must  lie  on  the 
circle  described  upon  E8  as  a  diameter.  This  will  be  apparent,  since 
XSIS  and  RSIS  are  at  right  angles.  The  phase  of  the  secondary  current 
is  the  same  as  that  of  the  vector  R8IS. 

The  primary  current  will  be  equal  to  the  secondary  current,  plus 


28  THE  INDUCTION  MOTOR 

the  no-load  current;  the  addition  is  of  course  to  be  made  vectorially. 
The  no-load  current  is  dependent  both  in  phase  and  magnitude  upon  the 
secondary  or  rotor  e.m.f.  Eg.  The  no-load  current,  taken  in  this  way,  is 
not  quite  the  same  as  what  we  ordinarily  call  by  this  name,  since  it  is 
necessarily  measured  before  it  enters  the  primary  or  stator,  and  conse- 
quently contains  a  power  component  due  to  the  stator  copper  loss.  The 
difference  is,  however,  negligible. 

If,  then,  for  any  given  secondary  e.m.f.  we  know  the  power  lost  in 
hysteresis  and  eddy  currents,  and  the  necessary  magnetizing  current, 
we  can  determine  the  components  of  the  no-load  current  and  lay  it  off  in 
proper  position  and  magnitude  as  shown  in  the  diagram.  The  primary 
or  stator  current  Ip  is  then  given  by  the  sum  of  I8  and  IQ.  It  is  now 
simple  to  find  the  primary  applied  e.m.f.  by  adding  to  the  secondary 
e.m.f.  Eg  the  vectors  XPIP  and  RPIP,  representing  respectively  the  react- 
ance drop  and  the  resistance  drop. 

The  difficulty  of  constructing  this  diagram  will  be  apparent  when  we 
consider  that  we  have  to  start  with  the  voltage  applied  to  the  secondary, 
and  this  is  not  known  until  the  diagram  is  completed,  since  the  applied 
voltage,  EP,  is  constant,  while  the  secondary  e.m.f.,  Es,  continually 
decreases  as  the  current  is  increased.  It  might  seem  that  we  could  con- 
struct the  diagram  as  shown,  determine  the  ratio  of  Ep  and  Eg  and  their 
difference  in  phase,  and  then  reconstruct  the  diagram,  drawing  E8  to  the 
proper  scale  to  start  with.  This  would  be  correct,  were  it  not  for  the 
fact  that  the  magnetizing  current  i$  is  not  proportional  to  the  secondary 
voltage.  Its  power  component  increases  about  in  proportion  to  the  1.8 
power  of  the  voltage,  and  its  wattless  component  somewhat  faster  than 
the  first  power  of  the  voltage,  the  rate  of  increase  being  greater  as  the 
saturation  of  the  iron  increases.  We  are  therefore  forced  to  the  con- 
clusion that  the  best  method  of  procedure  would  be  to  adopt  a  method 
of  trial  and  error,  each  new  attempt  giving  us  data  so  that  the  next 
attempt  would  be  more  nearly  correct. 

One  advantage  of  the  diagram  of  Fig.  19  is  that  if  desirable  we  may 
include  in  the  primary  resistance  and  reactance  the  resistance  and  react- 
ance of  the  lines  supplying  the  motor.  Thus  we  might  investigate  the 
action  of  the  motor  when  supplied  through  a  transmission  line,  the 
voltage  at  the  beginning  of  the  line  being  kept  constant. 

That  the  simpler  diagram  of  Fig.  1 7  is  sufficiently  accurate  and  that 
it  is  in  general  undesirable  to  use  the  more  complicated  ones,  particularly 
in  the  case  of  the  induction  motor,  will  be  more  apparent  when  we  con- 
sider that  all  of  these  diagrams  are  based  upon  the  waves  of  e.m.f.  and 


THEORY  OF  THE  INDUCTION  MOTOR  29 

flux  being  sinusoidal.  In  the  actual  motor,  with  an  applied  sine  wave  of 
e.m.f.,  none  of  the  other  waves  are  strictly  sinusoidal.  With  the  motor 
well  loaded,  the  stator  current  is  nearly  sinusoidal,  but  the  no-load  cur- 
rent, and  still  more,  the  rotor  current  are  badly  distorted.  It  will  be 
seen  that  on  this  account  the  theory  is  by  no  means  perfect,  and  an 
attempt  to  introduce  great  refinement  in  our  diagrams  is  inadvisable. 


CHAPTER  III 
STARTING  TORQUE 

IT  has  been  shown  in  a  previous  chapter  that  the  starting  torque  of 
an  induction  motor  may  be  expressed  in  synchronous  watts,  and  it 
has  been  shown  that  the  starting  torque  in  synchronous  watts  is  equal 
to  the  loss  in  the  rotor.  It  is  the  intention  in  the  present  chapter  to 
give  a  simple  proof  of  this  statement,  and  also  to  show  that  the  same 
rule  applies  to  the  starting  torque  of  any  motor  in  which  the  field  retains 
a  definite  form  with  respect  to  the  rotor.  The  demonstration  also 
applies  to  the  case  of  the  single-phase  commutator  type  motor,  in  fact 
to  any  of  the  commercial  types  of  motor  with  the  exception  of  the 
single-phase  induction  motor. 

To  apply  this  rule  we  must  understand  by  the  "  synchronous  "  speed 
of  a  machine,  the  speed  which  it  would  attain  if  it  were  allowed  to  run 
freely  without  friction  and  with  the  same  field  flux  as  that  used  in  start- 
ing. By  the  "  synchronous  watts  "  we  mean  the  power  the  machine 
would  deliver  if  it  were  operated  at  the  synchronous  speed  and  exerted 
the  given  starting  torque.  It  might  be  considered  as  the  torque  corre- 
sponding to  a  given  output  in  watts  at  the  synchronous  speed.  In  a 
direct-current  shunt  motor  at  the  instant  of  starting,  let  us  consider  that 
full-load  current  is  in  the  rotor.  The  machine  will  exert  full-load 
torque,  since  the  torque  in  this  case  is  obviously  independent  of  the  speed. 
The  power  expended  in  the  armature  and  the  resistance  in  series  with  it, 
whether  it  is  at  rest  or  in  motion,  is  equal  to  El.  If  the  machine  oper- 
ates at  full  load  and  100  per  cent  armature  efficiency  (which  would  be  the 
case  if  it  ran  at  the  synchronous  speed),  the  output  in  watts  is  the  same  as 
the  input  or  it  is  El.  Hence  the  starting  torque  in  synchronous  watts  is 
equal  to  the  power  expended  in  the  armature  circuit.  This  same  rule 
also  obviously  applies  irrespective  of  the  speed  of  the  rotor,  or  in  general 
the  torque  exerted  by  the  motor  in  synchronous  watts  is  equal  to  the 
power  expended  in  the  rotor  circuit. 

To  obtain  the  useful  torque  at  the  shaft  it  is  necessary  to  decrease 
the  torque  as  obtained  above  by  the  torque  required  to  overcome  the 

30 


STARTING  TORQUE  31 

mechanical  friction  of  the  motor,  and  that  required  to  overcome  the 
torque  due  to  hysteresis.  In  general  these  corrections  are  of  small  mag- 
nitude. 

In  the  case  of  the  series  motor,  it  is  evident  that  the  same  argument 
will  apply.  We  have  here,  however,  a  somewhat  different  case,  since  the 
syn  hronous  speed,  as  above  denned,  may  be  either  more  or  less  than  the 
full-load  speed.  If  less  than  full-load  current  is  passed  through  the 
motor  at  starting,  the  field  will  be  weak,  and  the  "  synchronous  speed  " 
as  above  defined  will  be  greater  than  the  full-load  speed.  If  the  starting 
current  be  greater  than  the  full-load  current,  the  "  synchronous  speed  " 
will  be  less  than  full-load  speed.  The  starting  torque  is  represented  by 
exactly  the  same  expression  as  before.  The  series  motor  has  therefore 
no  advantage  over  the  shunt  motor  in  regard  to  starting  torque,  provided 
the  starting  current  is  the  same  as  the  running  current.  This  is  contrary 
to  the  general  impression.  The  misconception  arises  from  neglecting 
the  consideration  of  the  final  speed  arrived  at  by  the  motor.  To  consider 
a  specific  case,  suppose  a  load  offering  a  constant  torque  of  500  ft-lbs.  is 
to  be  started  and  accelerated  to  1050  rev.  per  min.  Suppose  further 
that  this  torque  at  this  speed  represents  the  full-load  of  the  motor,  or  in 
this  case  100  h.p.  If  current  is  supplied  at  250  volts,  the  shunt  motor 
will  require  about  332  amperes  to  start  the  load.  The  series  motor  will 
require  slightly  less,  say  325  amperes,  since  there  is  no  shunt  field  to  be 
supplied.  The  one  motor  has  then  practically  no  advantage  over  the 
other. 

Suppose  the  torque  to  be  increased  to  750  Ibs.  The  shunt  motor  will 
require  approximately  50  per  cent  more  current  to  start,  say  498  amperes. 
Approximately  the  same  speed  as  before  will  be  attained,  and  the  output 
will  be  nearly  150  h.p.  The  series  motor,  on  the  other  hand,  will  require 
only  about  400  amperes  to  start,  and  attains  a  speed  of  852  rev.  per  min. 
This  is  on  the  supposition  that  the  fields  are  unsaturated.  The  h.p.  out- 
put will  consequently  be  122  h.p.  The  input  of  400  amperes  at  250  volts 
is,  allowing  10  per  cent  for  losses,  just  equal  to  122  h.p. 

The  case  of  the  series  single-phase  motor  is  readily  seen  to  be  the 
same  as  that  of  the  series  direct-current  motor. 

The  starting  torque  of  the  induction  motor  is  of  the  greatest  interest 
to  us  here.  The  facts  in  this  case  will  perhaps  be  most  readily  seen  if 
we  consider  the  rotor  of  an  induction  motor  to  be  at  rest,  and  that  a  field 
magnet,  excited  by  direct  current,  is  rotated  around  it.  The  conditions 
will  be  the  same  as  those  in  an  induction  motor  during  the  starting 
period,  providing  the  rotating  magnetic  field  in  the  motor  is  constant  in 


32  THE  INDUCTION  MOTOR 

strength,  rotates  at  a  uniform  velocity,  and  preserves  a  uniform  distribu- 
tion. In  the  case  of  the  rotating  field,  a  certain  torque  will  be  required 
to  maintain  the  motion.  Neglecting  the  friction  losses  of  the  field,  it  is 
evident  that  the  same  torque  will  be  exerted  upon  the  rotor  as  upon  the 
field.  All  the  power  applied  with  the  exception  of  the  small  friction  loss 
must  be  dissipated  in  the  rotor.  It  is  evident  at  once  that  the  torque  in 
synchronous  watts  is  equal  to  the  loss  in  the  rotor.  The  case  is  the  same 
in  an  induction  motor,  provided  the  conditions  just  mentioned  are  com- 
plied with.  This  is  nearly  the  case  in  practice. 

The  above  obviously  applied  both  to  the  squirrel-cage  and  to  the 
wound-rotor  type  of  machine.  It  would  seem  then,  that  the  starting 
would  be  equally  efficient  for  either  one.  The  difficulty  with  the  squirrel- 
cage  type  arises,  not  because  more  power  must  be  supplied  to  the  rotor, 
but  on  account  of  the  difficulty  of  getting  it  there.  In  the  first  place  the 
copper  loss  in  the  rotor  in  normal  operation,  in  a  typical  modem  squirrel- 
cage  motor,  will  be  perhaps  twice  that  in  the  stator.  Hence  in  addition  to 
the  rotor  loss,  we  have  a  loss  of  about  half  as  much  in  the  stator.  This 
corresponds  to  the  field  loss  in  a  shunt  motor,  but  whereas  the  field  loss 
in  the  shunt  machine  is  very  small,  here  it  is  a  large  proportion  of  the 
rotor  loss.  In  fact  in  squirrel-cage  motors  built  for  small  slip  and  conse- 
quently small  starting  torque,  the  stator  loss  may  be  as  much  as  twice  the 
rotor  loss.  This  is  avoided  in  the  case  of  the  wound-rotor  machine,  by 
causing  the  loss  in  the  rotor  to  be  temporarily  several  times  as  largeas  the 
stator  loss. 

The  other  difficulty  is  that  on  account  of  the  inductance  of  the  induc- 
tion-motor windings,  the  current  is  supplied  at  a  very  low  power-factor. 
This  means  that  a  large  current  must  be  taken  from  the  line,  and  that  on 
account  of  the  low  power-factor,  a  great  disturbance  of  voltage  is  pro- 
duced. We  may  put  the  matter  this  way.  Neglecting  the  field  losses, 
the  series  motor,  either  for  direct  or  alternating  current,  the  shunt  motor, 
and  the  induction  motor  all  require  the  same  amount  of  power  in  starting. 
This  power  is  the  same  in  watts  as  the  starting  torque  in  synchronous 
watts.  In  the  series  motor,  the  field  loss  is  zero  (since  the  resistance  of 
the  field  takes  the  place  of  resistance  which  would  otherwise  have  to  be 
supplied).  In  the  shunt  motor  the  field  loss  is  small,  being  from  3  per 
cent  to  i  per  cent  or  less.  In  the  case  of  the  induction  motor,  this  loss  is 
also  small  if  a  wound-rotor  machine  is  used,  and  is  about  the  same  as  in  a 
shunt  motor  of  corresponding  size.  In  the  squirrel-cage  motor,  however, 
this  field  loss  is  considerable,  varying  from  30  per  cent  to  300  per  cent  or 
more  of  the  rotor  loss. 


STARTING  TORQUE  33 

The  above  is  on  the  supposition  that  the  shunt  and  series  motors  are 
supplied  with  current  at  a  constant  voltage,  and  that  the  induction  motor 
is  supplied  with  current  at  just  the  voltage  to  force  the  required  current 
through  the  windings.  This  is  done  at  least  approximately  in  practice 
in  the  case  of  squirrel-cage  induction  motors  by  using  an  auto-trans- 
former to  give  the  correct  voltage.  If  this  could  be  done  in  the  case  of  the 
shunt  and  series  direct-current  motors,  their  starting  efficiency  would  of 
course  greatly  exceed  that  of  the  induction  motor.  Likewise  in  the 
induction  motor  with  wound  rotor,  if  the  current  produced  in  the  second- 
ary could  be  utilized,  the  starting  efficiency  would  be  very  greatly 
increased.  To  do  this,  however,  would  be  very  difficult,  since  during 
the  process  of  starting,  the  power  would  be  supplied  at  both  varying 
voltage  and  varying  frequency.  No  practicable  method  of  doing  this  is 
available  at  the  present  time  unless  we  regard  the  case  of  two  motors 
operating  in  cascade  as  being  an  exception.  The  reader  should  of  course 
bear  in  mind  that  the  production  of  torque  alone  requires  theoretically 
no  power.  In  practice,  the  only  power  required,  provided  advantage  is 
taken  of  the  methods  above  pointed  out,  is  that  required  to  supply  the 
copper  losses  and  in  the  case  of  the  induction  motor,  the  iron  loss. 

STARTING  TORQUE  OF  COMMERCIAL  SQUIRREL-CAGE  INDUCTION 
MOTORS 

The  question  of  the  design  of  the  squirrel-cage  induction  motor  so  as 
to  give  suitable  starting  torque  is  a  very  serious  one.  The  designer  has 
to  choose  between  high  starting  torque  and  high  efficiency.  A  slightly 
different  way  of  expressing  the  starting  torque  may  help  to  make  this 
clearer.  As  was  pointed  out,  the  starting  torque  in  synchronous  watts 
is  equal  to  the  loss  in  the  rotor.  Suppose  we  have  full-load  current 
hrough  the  stator  at  start 'ng.  The  rotor  current  will  be  approximately 
the  same  as  the  rotor  current  at  full  load,  and  consequently  the  rotor  loss 
will  be  equal  to  the  full-load  rotor  loss.  The  slip  is  equal  to  the  rotor 
loss  divided  by  the  input  o  the  rotor.  Likewise  the  starting  torque  in 
percentage  of  the  full-load  starting  torque  is  equal  to  the  same  quantity. 
Consequently  the  starting  torque  with  full-load  current  is  equal  to  the 
slip.  For  example,  a  motor  with  a  slip  of  4  per  cent  will  have  a  starting 
torque  of  only  4  per  cent  when  the  motor  takes  full-load  current.  The 
(orque  increases,  however,  in  proportion  to  the  square  of  the  current. 
An  average  motor  will  take  about  six  times  full-load  current  if  thrown 
directly  across  the  line.  Consequently  the  motor  considered  would 


34 


THE  INDUCTION  MOTOR 


develop  a  maximum  starting  torque  of  62X4=i44  per  cent.  This 
neglects  the  effect  of  the  iron  losses  in  increasing  the  torque.  This 
increase  will  be  treated  later.  The  expression  for  the  percentage  of 


starting  torque  may  then  be  written  thus:  D=  {  —  }  s,  where  Ic  is  the  cur- 

Vi/ 

rent  per  phase  taken  when  the  motor  is  thrown  direct  y  across  the  line, 
/i  is  the  full-load  current,  and  s  is  the  slip.  Thus  to  increase  the  starting 
torque  we  can  do  either  one  of  two  things:  increase  the  slip,  or  increase 
the  starting  current.  To  do  the  first  is  obviously  undesirable,  since  it 
decreases  the  efficiency.  It  is  likewise  undesirable  to  increase  the  start- 
ing current  beyond  a  certain  value.  The  most  obvious  objection  is  that 
to  do  so  impairs  the  regulation  of  the  line,  demands  larger  auto-trans- 


FIG.  20. — Comparative  Circle  Diagrams  of  Induction  Motor  with  Few  and  Many 
Turns  per  Coil. 


formers  or  rheostats  for  starting,  and  is  apt  to  cause  excessive  heating 
of  the  windings  of  both  stator  and  rotor. 

Of  even  more  importance  is  the  fact  that  increasing  the  starting  cur- 
rent to  too  great  a  value  makes  the  characteristics  of  the  motor  very 
poor.  This  is  particularly  true  of  the  power-factor.  This  will  be  readily 
seen  from  Fig.  20.  A  represents  the  circle  diagram  of  a  motor  in  which 
the  starting  current  is  five  times  the  full-load  running  current.  The  idle 
current  is  taken  in  both  cases  equal  to  one-fourth  the  full-load  current. 
In  curve  B  the  starting  current  is  taken  as  eight  times  the  full-load  cur- 
rent. C  and  D  represents  respectively  the  points  on  the  two  circles  cor- 
responding to  full  load.  Since  the  cosines  of  the  angles  COE  and  DOE 
represent  respectively  the  power-factors  in  the  two  cases,  it  is  easily  seen 
that  the  power-factor  of  the  motor  A,  at  full  load,  is  much  better  than  that 
of  B.  In  fact  the  power-factor  of  A  is  better  for  all  loads  from  zero  to  1 75 


STARTING  TORQUE  35 

per  cent  of  full  load.  If  the  slip  in  both  cases  is  4  per  cent  the  motor  A 
will  develop  52X4=iooper  cent  and  the  motor  B  L2X4=2$6  per  cent 
starting  torque.  However,  the  motor  B  is  so  much  inferior  in  power- 
factor  throughout  all  the  operating  range,  as  to  make  it  highly  undesir- 
able. Moreover,  if  the  load  to  be  started  requires  a  greater  starting 
torque  than  100  per  cent,  we  should  get  equally  as  good  results  by 
using  motor  A  and  stepping  up  the  voltage  for  starting.  For  any  given 
torque,  say  150  per  cent,  the  current  required,  and  the  power-factor 
would  be  the  same  in  the  two  cases. 

Before  leaving  this  subject,  it  should  be  pointed  out  that  the  employ- 
ment of  a  large  starting  current  to  secure  large  starting  torque  is,  in  gen- 
eral, more  justified  in  the  case  of  25-cycle  motors  than  in  the  case  of  60- 
cycle  motors.  This  is  due  to  the  fact  that  on  account  of  the  smaller 
number  of  poles,  the  power-factor  in  25-cycle  motors  is  inherently  higher 
than  in  the  case  of  the  higher  frequency;  hence,  the  lowered  power- 
factor  is  not  so  objectionable. 

If,  on  the  other  hand,  we  attempt  to  secure  large  starting  torque  by 
increasing  the  slip,  we  reduce  greatly  the  efficiency,  and  impair  the  speed 
regulation  of  the  motor.  Just  what  this  means  in  dollars  and  cents  may 
be  shown  by  the  following  typical  case.  Consider  a  loo-h.p.  motor  for 
which  it  is  specified  that  the  starting  torque  must  be  at  least  175  per  cent 
with  a  current  not  exceeding  5.5  times  full -load  current.  This  is  by  no 
means  an  unusual  requirement.  The  slip  will  be  175  -^5-52=  5.72  per 
cent.  When  the  motor  is  operating  at  full  load,  this  means  a  constant 
rotor  loss  of  approximately  6  h.p.  or  4^  k.w.  If  the  motor  is  operated  six 
hours  per  day,  300  days  per  year,  and  energy  is  sold  at  the  low  rate  of  2 
cents  per  k.w.-hr.,  the  cost  of  the  energy  lost1  in  the  rotor  yearly  is  $162. 
At  least  half  of  this  could  be  saved  by  constructing  the  machine  with  a 
wound  rotor  of  low  resistance.  This  saving  would  justify  an  investment 
of  $547,  taking  interest  and  depreciation  at  the  high  value  of  15 
per  cent.  This  would  be  ample  to  cover  the  extra  cost  of  a  wound-rotor 
machine  instead  of  the  squirrel  cage,  or  of  a  loose  pulley  and  friction 
clutch  on  the  counter  shaft.  This  is  a  point  which  is  frequently  not 
given  the  consideration  which  its  importance  demands.  A  little  effort 
to  get  easy  starting  conditions,  and  the  use  of  a  motor  with  small  slip, 
will  pay  large  dividends  upon  the  time  and  money  invested. 


36  THE  INDUCTION  MOTOR 

COMPARATIVE  STARTING  TORQUES  OF  SQUIRREL-CAGE  AND  WOUND- 
ROTOR  MOTORS 

It  is  a  general  impression  that  the  induction  motor  with  wound  rotor 
is  far  superior  in  the  matter  of  starting  torque  to  the  squirrel-cage 
machine.  This  is  true  to  a  certain  extent,  but  to  a  far  less  degree  than  is 
generally  supposed. 

To  take  atypical  case  assume  a  2o-h.p.,  6o-cycle,  i2oo-rev.  per.  min. 
three-phase  machine.  This  in  the  squirrel-cage  type  would  develop  a 
starting  torque  of  about  200  per  cent,  taking  a  current  of  5  ^  times  full- 
load  current  to  do  so.  To  start  under  full-load  torque  it  would  require 
70.7  per  cent  of  full  voltage  to  be  applied  to  it.  The  motor  current  would 
then  be  70.7  per  cent  of  5.5  or  3.89  times  full-load  current.  If  we  assume 
an  auto-transformer  to  be  used  and  allow  nothing  for  the  loss  in  it,  the 
line  current  will  be  one-half  of  5.5  or  2.75  times  full-load  current.  To 
develop  this  torque  will  require  an  expenditure  of  20  h.p.  in  the  rotor.  If 
the  full-load  copper  loss  in  the  stator  is  4  per  cent  we  shall  have  during 
starting  a  loss  of  4  X3-892=  60  per  cent,  and  if  the  iron  loss  at  full  voltage 
is  5  per  cent,  the  iron  loss  during  starting  will  be  about  3  per  cent.  This 
is  true  since  the  iron  loss  decreases  somewhat  faster  than  the  voltage. 
The  total  input  will  then  be  about  163  per  cent  of  20  h.p.,  or  32.6  h.p.  To 
this  something  should  be  added  on  account  of  the  loss  in  the  auto- 
starter. 

In  the  case  of  the  wound-rotor  machine,  the  starting  current  for  100 
per  cent  torque  would  be  approximately  equal  to  the  full-load  current. 
The  full  line  voltage  would  of  course  be  applied.  As  before,  we  should 
have  to  expend  20  h.p.  in  the  rotor.  In  addition  we  should  have  a  loss  of 
4  per  cent  in  the  stator  copper  and  5  per  cent  in  the  stator  iron  loss.  The 
total  input  is  then  109  per  cent  of  20  h.p.  or  21.8  h.p. 

Comparing  the  two,  it  will  be  seen  that  the  wound-rotor  machine  is 
somewhat  superior  in  that  it  takes  only  21.8  h.p.  compared  wi'.h  32.6  for 
its  rival.  The  cost  of  the  energy  used  in  starting  is  therefore  somewhat 
less.  In  the  matter  of  current  required  it  is  decidedly  superior,  since  it 
requires  only  one,  compared  with  2f  times  full-load  current.  Moreover 
the  power-factor  in  the  case  of  the  wound-rotor  machine  is  about  85  per 
cent  compared  with  35  per  cent  in  the  case  of  the  other.  If  the  motor  is 
operated  upon  the  circuits  of  a  public  service  company,  this  is  no  great 
disadvantage  to  the  user  of  the  motor.  It  is,  however,  a  decided  one 
from  the  standpoint  of  the  company,  on  account  of  the  deleterious  effect 
of  low  power-factor  upon  the  regulation  of  the  circuit.  If  the  motor  is  of 


STARTING  TORQUE  37 

a  size  comparable  with  the  generator  supplying  the  energy,  it  may  be  a 
very  serious  matter. 

For  example,  consider  the  case  of  a  ico-h.p.  motor  supplied  from  a 
100  k.w.  generator.  Such  a  motor  would  require  120  amperes  when 
operating  under  full  load  on  a  440-volt,  three-phase  circuit.  During 
starting,  however,  it  would  require,  as  we  have  shown,  about  2§  times  as 
much  current  or  330  amperes.  This  is  on  the  assumption  that  the  motor 
is  developing  100  per  cent  torque.  If  the  generator  operates  at  460  volts 
to  allow  for  drop  in  line,  its  full-load  current  will  be  109  amperes.  The 
generator  has  then  during  the  starting  of  the  motor  a  power  load  of  133 
h.p.,  plus  the  loss  in  transmission  line,  or  say  104  k.w.,  and  an  apparent 
load  of  263  k.v.-amp.  The  power-factor  of  the  load  is  39.5  per  cent. 
It  is  unlikely  that  under  this  large  load  the  generator  would  be  able  to 
keep  up  its  voltage,  and  it  would  probably  be  impossible  to  start  the 
motor:  Such  a  combination  would  work  only  in  case  the  motor  could  be 
started  with  a  very  light  load.  If,  on  the  other  hand,  the  motor  is  of  the 
wound-rotor  type,  the  power  load  will  be  81  k.w.  and  the  apparent  load 
will  be  92  k.v.-amp.  The  power-factor  will  consequently  be  88  per  cent, 
and  no  difficulty  should  be  experienced  in  starting  under  full  load. 

INFLUENCE  or  POOR  LEAKAGE  COEFFICIENT,  IN  WOUND-ROTOR 
MACHINES 

To  one  who  has  studied  most  of  the  text-books  upon  induction  motors, 
it  is  rather  a  surprise  to  find  that  in  many  cases  a  squirrel-cage  motor  is 
guaranteed  to  develop  the  same  maximum  torque  as  the  corresponding 
wound-rotor  machine  of  the  same  maker.  This  fact  is  partially  explained 
by  the  fact  that  the  torque  is  more  irregular  in  a  motor  of  the  latter  type. 
A  factor  of  more  importance  is,  however,  the  fact  that  the  wound-rotor 
machine  has  a  much  larger  leakage  coefficient  than  the  squirrel-cage 
machine.  This  is  due  to  the  longer  end  connections  in  the  former,  and 
to  the  fact  that  the  currents  are  confined  to  definite  paths  instead  of  being 
allowed  to  pick  their  own  path. 

On  account  of  the  advisability  of  using  standard  windings,  the  same 
stator  is  usually  used  for  a  machine  of  a  given  rating,  irrespective  of  the 
type  of  rotor.  As  a  wound-rotor  machine,  the  leakage  coefficient  will  be 
approximately  one-third  greater  than  in  the  case  of  the  squirrel-cage 
machine.  Hence  the  circle  diagram  of  the  latter  machine  will  be  33  per 
cent  larger  in  diameter,  and  the  current  in  the  rotor  at  start  33  per  cent 
greater.  Since  the  starting  torque  is  proportional  to  the  square  of  the , 


38  THE  INDUCTION  MOTOR 

current,  the  torque  with  the  same  equivalent  rotor  resistance  would  be 
1.78  times  as  great.  Of  course  the  rotor  resistance  of  the  wound-rotor 
machine  would  hi  practice  be  made  much  more  than  that  of  the  squirrel- 
cage  machine,  and  this  would  usually  be  more  than  enough  to  counteract 
the  better  leakage  coefficient.  However,  the  other  facts  mentioned,  such 
as  lack  of  uniformity  of  the  torque,  would  act  to  cut  down  this  advantage, 
and,  as  previously  stated,  the  torque  of  the  squirrel-cage  machine  will  in 
many  cases  be  found  to  be  fully  as  high  as  that  of  its  rival.  It  should  also 
be  noted  that  on  account  of  the  lower  leakage  coefficient,  the  power- 
factor  and  the  pull-out  point  of  the  squirrel-cage  machine  will  be  materi- 
ally higher  than  in  the  case  of  the  wound-rotor  machine.  Rewinding  the 
machine  with  a  smaller  number  of  turns  will,  it  is  true,  remedy  the  latter 
fault,  but  it  leads  to  a  large  iron  loss,  and  the  power-factor  will  generally 
be  somewhat  poorer.  On  account  of  its  inherently  better  characteris- 
tics, the  squirrel-cage  machine  should  be  used  wherever  possible;  The 
rotor  should,  however,  be  preferably  of  low  resistance,  so  as  not  to  offset 
the  better  power-factor  and  pull-out  point  by  a  lowered  efficiency. 

EFFECT  OF  IRON  LOSSES 

We  have  seen  that  in  order  to  develop  any  given  torque,  it  is  neces- 
sary that  there  be  a  loss  in  the  rotor  equal  to  the  power  that  would  be 
developed  by  the  motor  if  running  at  synchronous  speed  with  the  given 
torque.  It  will  be  at  once  evident  that  it  does  not  matter  whether  this 
loss  is  copper  loss  or  iron  loss.  As  a  general  thing,  the  iron  loss  is  not  of 
much  assistance  in  increasing  the  starting  torque,  but  in  some  cases  it 
may  make  a  decided  difference.  The  writer  has  seen  a  number  of  cases 
in  which  the  starting  torque  was  increased  at  least  50  per  cent  by  the  iron 
loss. 

To  increase  the  torque  in  this  wray  is  decidedly  advantageous,  since 
this  rotor  iron  loss  disappears  almost  entirely,  when  the  motor  is  in  nor- 
mal operation.  This  is  due  to  the  fact  that  the  frequency  in  the  rotor  is 
very  low  when  the  motor  is  operating  near  synchronous  speed.  Conse- 
quently the  rotor  iron  loss  is  available  when  wanted  for  starting,  and 
disappears  when  the  motor  is  up  to  speed  and  it  is  no  longer  needed. 
The  iron  loss  mentioned  is  that  normally  due  to  the  pulsation  of  the  flux 
hi  the  rotor,  and  should  be  carefully  distinguished  from  the  excessive 
loss  sometimes  found  in  the  rotor  teeth  at  or  near  synchronism.  This 
phenomenon  is  treated  in  detail  elsewhere.  The  effect  of  this  if  present 
is  decidedly  disadvantageous,  since  it  is  greatest  near  synchronism,  and 
disappears  at  zero  speed. 


STARTING  TORQUE  39 

FLUCTUATION  OF  TORQUE  IN  WOUND-ROTOR  MACHINES 

The  case  of  the  wound-rotor  machine  is  somewhat  less  favorable  than 
as  above  presented,  since  we  have  neglected  the  fact  that  the  torque  of  a 
machine  of  this  type  is  by  no  means  constant,  but'varies  widely  in  differ- 
ent positions  of  the  rotor  with  respect  to  the  stator.  This  fluctuation  is 
present  to  some  extent  in  the  squirrel-cage  motor,  but  is  usually  of  negli- 
gible importance.  It  is  greater,  the  greater  the  current  and  the  fewer 
the  number  of  phases  in  the  rotor.  This  is  largely  the  reason  for  the 
superiority  of  the  squirrel-cage  machine,  since  here  the  number  of  phases 
is  very  great. 

If  a  wound-rotor  machine  be  started  under  a  small  torque,  say  50  per 
cent  the  fluctuation  may  amount  to  perhaps  25  per  cent  of  the  average^' 
torque.  If  full-load  torque  is  developed,  the  fluctuation  may  be  as  much 
as  70  per  cent.  If  we  go  to  the  limit  and  short-circuit  the  rotor,  it  will  be 
found  that  the  fluctuation  is  so  great  that  the  torque  frequently  drops  to 
zero  in  certain  positions,  or  even  that  the  rotor  locks  in  these  places,  and 
torque  must  be  applied  to  get  the  rotor  past  the  point,  or  the  current  must 
be  shut  off  and  a  fresh  start  made.  The  rotor  is  of  course  not  intended 
to  be  short-circircuited  during  the  starting  period,  but  it  may  easily 
happen  when  starting  a  heavy  load  with  an  automatic  starter. 

Various  expedients  are  employed  to  reduce  this  fluctuation.  Some 
of  these  are  the  use  of  short-pitch  windings  on  the  stator  and  rotor,  mak- 
ing the  rotor  winding  unsymmetrical,  and  the  use  of  spiral  slots  in  the 
rotor,  produced  by  placing  each  rotor  lamination  at  a  slight  angle  to  the 
one  before  it.  By  the  use  of  one  or  more  of  these  devices,  it  is  possible  to 
reduce  the  fluctuation  to  20  per  cent  or  less,  at  full-load  torque. 

The  reason  for  this  fluctuation  in  the  torque  will  be  apparent  if  we 
consider  that,  as  shown  on  page  30,  the  torque  in  synchronous  watts  is 
equal  to  the  loss  in  the  rotor,  only  on  condition  that  the  flux  does  not 
change  either  in  its  distribution  or  in  value  during  rotation.  Thus  con- 
sidering the  case  considered  there,  that  of  a  rotating  field  magnet  revolved 
around  the  rotor  of  an  induction  motor,  it  was  shown  that  if  the  field  did 
not  vary  all  of  the  power  imparted  to  the  stator  appeared  in  the  rotor  as 
I2R  loss,  and  since  the  torque  on  the  field  and  that  on  the  rotor  are  equal, 
the  torque  in  synchronous  watts  is  equal  the  loss  in  the  rotor. 

If,  on  the  other  hand,  we  should  keep  the  field  at  rest,  and  excite  it  by 
means  of  an  alternating  current,  it  is  evident  that  we  should  have  a  large 
current  produced  in  the  rotor  and  in  consequence  a  large  loss  in  it,  but 
no  torque  at  all  would  be  developed.  We  have  seen  that  in  the  case  of 


40  THE  INDUCTION  MOTOR 

a  motor  operating  near  synchronism,  the  rotor  acts  powerfully  to  prevent 
any  change  in  the  shape  of  the  flux  distribution  wave.  At  standstill  this 
action  is  nearly  absent,  and  in  consequence,  even  with  a  sine  wave  of 
applied  e.m.f .,  the  flux  wave  may  be  very  much  distorted,  and  what  is 
more  serious,  may  change  materially  in  its  space  distribution  from  point 
to  point.  This  action  is  seriously  increased  from  the  fact  that  as  the  rotor 
revolves,  the  relative  positions  of  the  stator  and  rotor  coils  change,  and 
in  consequence  the  impedance  of  the  motor  is  changing.  An  ammeter 
or  wattmeter  connected  in  the  primary  circuit  will  show  the  fluctuation  of 
the  current  or  power.  This  last  action  is  almost  entirely  absent  in  a 
well-designed  squirrel-cage  motor.  The  fluctuation  in  the  shape  of  the 
flux  wave  is,  however,  present  to  some  extent  in  both  the  squirrel-cage 
and  in  the  wound-rotor  machine. 

It  will  be  readily  seen  from  the  above  that  when  a  motor  is  in  the  act 
of  starting,  there  exists  a  rotating  magnetic  flux  which  is,  however, 
neither  of  constant  value  or  of  constant  space  distribution.  We  may 
consider  roughly  that  that  portion  of  the  flux  which  does  not  change  acts 
to  produce  the  torque  of  the  motor.  The  variable  portion  sets  up  cur- 
rents in  the  rotor  which  are  largely  ineffective  in  producing  torque.  In 
the  case  of  the  wound-rotor  machine,  we  have  the  further  fact  that  the 
torque  in  addition  to  being  weakened  by  the  process  just  described,  varies 
at  different  points  of  the  revolution,  due  to  the  varying  relative  positions 
of  the  rotor  and  stator. 

As  is  pointed  out  elsewhere,  the  use  of  a  fractional  pitch  winding  is 
equivalent  to  allowing  the  different  bands  of  current  to  overlap.  This 
has  the  effect  of  making  the  fluctuation  of  the  rotating  flux  at  starting 
less,  and  in  consequence  the  starting  torque  will  be  increased.  It  might 
also  be  pointed  out  that  the  single-phase  induction  motor  represents 
the  extreme  case  of  fluctuation  of  the  flux.  In  fact,  at  the  moment  of 
starting,  there  is  no  rotating  component,  and  all  of  the  flux  may  be  con- 
sidered as  being  a  fluctuating  component.  The  starting  torque  is  there- 
fore zero. 


CHAPTER   IV        , 
STARTING  DEVICES 

IN  starting  an  induction  motor,  various  devices  are  used  to  reduce  the 
current  required.  In  the  case  of  the  wound-rotor  machine,  the  starting 
device  takes  the  form  of  a  three-phase  resistor  or  its  equivalent.  This 
usually  has  such  resistance  as  to  allow  approximately  full-load  current 
to  flow  when  it  is  all  in  circuit.  The  torque  developed  is  then  approxi- 
mately equal  to  the  full-load  torque  of  the  motor.  The  maximum  torque 
that  the  motor  can  develop  is  attained  when  the  value  of  this  resistance  is 
such  that  the  end  of  the  current  vector  is  at  nearly  the  top  of  the  circle. 
The  power-factor  of  the  rotor  current  is  then  evidently  equal  to  70.7  per 
cent,  since  the  angle  of  lag  is  45  degrees.  A  decrease  of  the  resistance 
below  the  value  required  to  give  this  current  would  evidently  cause  the 
motor  to  take  more  current  and  at  the  same  time  give  a  reduced  torque. 
Hence  it  would  never  be  advisable  to  reduce  the  resistance  on  the  first 
contact-point  below  this  value.  If  the  resistor  were  to  remain  in  circuit 
permanently  the  above  conclusion  would  of  course  require  modification, 
but  in  this  case  a  squirrel-cage  machine  would  be  preferable. 

The  value  of  resistance  to  produce  the  above  torque  can  be  very 
readily  determined  from  the  completed  machine.  To  do  this  it  is  merely 
necessary  to  connect  the  motor  with  a  wattmeter,  ammeter,  and  volt- 
meter in  circuit.  The  secondary  resistance  can  then  be  varied  until  the 
input  is  a  maximum.  This  will  be  nearly  the  point  of  maximum  torque 
as  is  at  once  apparent  from  the  circle  diagram.  Of  course  the  torque 
may  be  measured  directly  if  more  convenient.  It  is  also  evident  that  the 
value  of  the  resistance  may  be  calculated  directly,  if  the  circle  diagram 
of  the  motor  is  known. 

In  the  case  of  any  except  the  smallest  squirrel-cage  motor,  a  starting 
device  is  necessary,  not  so  much  on  account  of  the  motor,  but  on  account 
of  the  circuit  and  the  connected  machinery.  As  far  as  the  motor  is  con- 
cerned, it  would  not  be  injured  by  the  large  momentary  current.  In  any 
event  the  power  expended  in  bringing  the  load  up  to  speed  is  wasted. 
The  power  to  supply  this  loss  is  furnished  at  approximately  the  same 
efficiency  whether  the  load  is  accelerated  slowly  or  rapidly.  The  power 

41 


42  THE  INDUCTION  MOTOR 

wasted  in  friction,  on  the  other  hand,  will  be  greater  the  slower  the 
acceleration.  Hence  the  total  amount  of  heat  developed  in  the  motor 
will,  if  anything,  be  less  if  the  motor  be  thrown  directly  on  the  line. 

This  procedure  would,  however,  in  most  cases  result  in  drawing  such 
a  large  current  from  the  line  that  the  voltage  of  the  circuit  would  be  seri- 


FIG.  2i.— Western  Electric  Auto-starter. 

ously  lowered,  and  hence  it  would  be  objectionable.  In  many  cases  also, 
the  acceleration  would  be  so  rapid  that  the  belts  would  slip  and  be  dam- 
aged or  the  connected  machinery  might  be  injured.  The  point  to  be 
noted  is  that  the  starting  devices  are  necessary  on  account  of  external 
actors,  and  not  primarily  for  the  sake  of  the  motor. 

The  most  usual  form  of  starting  device  for  squirrel-cage  motors  is  an 
auto-transformer  to  reduce  the  voltage  applied  to  the  motor  terminals 
during  the  process  of  starting.  After  full  speed  is  attained  the  trans- 


STARTING  DEVICES  43 

former  is  disconnected  from  the  line.  Usually  the  voltage  is  applied  in 
two  steps,  a  reduced  voltage  and  the  full-line  pressure,  but  in  the  case  of 
some  very  large  motors  the  pressure  may  be  applied  in  several  steps. 
The  external  appearance  of  an  auto-starter  is  shown  in  Fig.  2 1 ,  and  the 
internal  appearance  in  Fig.  22. 

The  transformers  used  in  a  starter  of  this  type  are  usually  auto-trans- 


FIG.  22.— Western  Electric  Auto-starter.    Interior  View. 

formers.  The  amount  of  copper  required  in  an  auto-transformer  is 
much  less  than  would  be  required  in  one  having  both  primary  and  second- 
ary coils.  This  is  particularly  the  case  when  the  ratio  of  transformation 
is  near  to  one.  In  the  limiting  case,  when  the  ratio  is  one,  the  power 
rating  of  a  given  transformer  is  infinite.  The  truth  of  these  facts  can 
be  readily  seen  from  Fig.  23.  For  simplicity  it  has  been  assumed  that 
the  transformer  has  100  turns  in  all,  and  that  the  secondary  is  tapped  off 
from  70  turns.  If  then  the  primary  voltage  is  100,  the  secondary  voltage, 


44 


THE  INDUCTION  MOTOR 


To  Line  J 
100  Turns.  100  Volts  | 

l? 

loJopsT 

A 

.  100  Amns. 

. 

70  Turns.  70  Volts 
To  Motor 

52gr 

~~ 

~  • 

of     Auto- 

FIG.     23.  —  Representation 
transformer. 

neglecting  losses,  will  be  70.  Let  us  assume  that  the  current  taken  from  the 
secondary  is  100  amperes.  Since  the  secondary  output  is  70  X 100  =  7000 

volt-amperes,  the  primary  current 
must  be  70  amperes,  giving  the 
same  volt-ampere  input. 

The  currents  in  the  various  parts 
of  the  windings  are  readily  deduced. 
Obviously  the  current  in  the  part  A 
B  is  70  amperes.  Since  the  current 
taken  from  the  transformer  is  100 
amperes  and  since  it  is  the  sum  of 
the  currents  in  the  windings  A  B  and 
B  C,  the  current  in  the  latter  must 
be  30  amperes.  If  the  transformer 
were  provided  with  both  a  primary 
and  a  secondary  coil,  the  primary 
would  consist  of  100  turns  of  wire 

large  enough  to  carry  70  amperes,  and  the  secondary  of  70  turns  capable 
of  carrying  100  amperes.  Thus  the  amount  of  wire  required  for  the  auto- 
transformer  is  readily  seen  to  be  only  30  per  cent  as  much  as  would  be 
needed  for  the  type  having  two  coils.  In  fact  it  can  readily  be  seen  that 
the  amounts  of  wire  required  in  the  two  types  are  in  the  ratio  of  Ep  — E8  to 
Ep,  where  Ep  is  the  primary  voltage  and  E8  is  the  secondary  voltage. 
Since  for  starting  induction  motors  the  reduction  of  voltage  is  rarely 
more  than  50  per  cent,,  the  great  advantage  of  auto-transformers  is  readily 
comprehended. 

Moreover  an  induction-motor  starter  is  usually  provided  with  several 
taps,  so  that  a  number  of  secondary  voltages  can  be  obtained,  depending 
upon  the  load  to  be  started.  It  is  clear  that  at  the  larger  voltages  a 
greater  output  is  taken  from  the  transformer.  This  is  readily  provided 
for  in  the  auto-transformer,  since  the  power  capacity  is  much  greater  at 
the  larger  ratios  of  transformation.  Of  course  in  actually  designing  an 
auto-transformer,  not  all  of  the  saving  would  be  made  in  the  copper  as 
indicated,  but  the  section  of  iron,  as  well  as  that  of  the  copper,  would  be 
reduced.  Since  the  apparatus  is  for  intermittent  use  only,  both  the  cop- 
per and  the  iron  are  worked  at  very  high  densities,  and  the  apparatus  is 
very  small  compared  to  its  power  rating. 

For  two-phase  circuits,  two  auto-transformers  are  used,  one  connected 
across  each  of  the  two  phases.  In  the  case  of  a  three-phase  starter, 
there  are  two  types  in  general  use.  The  one  uses  a  three-phase  trans- 


STARTING  DEVICES 


45 


former,  as  shown  in  Fig.  24.  The  diagram  also  shows  the  connections 
of  the  starting  switch.  This  latter  is  usually  of  the  oil-immersed  type, 
and  is  generally  constructed  as  an  integral  part  of  the  starter.  The 
middle  row  of  contacts  are  mounted  on  a  drum,  and  can  be  thrown 
either  up  or  down  so  as  to  connect  to  either  the  upper  or  the  lower  row 
of  contacts.  The  connections  can  be  readily  traced  out  and  it  will 
be  seen  that  when  the  switch  is  in  the  upper  position  the  motor  is  con- 
nected directly  to  the  line  through  the  fuses,  and  the  transformer  is 
entirely  disconnected  from  the  line.  In  the  lower  starting  position 
of  the  switch,  the  transformers  are  in  circuit,  but  connected  outside  of 
the  fuses.  This  is  necessary  in  order  that  the  fuses  may  not  be  blown 

To    I    "Line 

5^*n 

'1!  I       1  To  Motor 


FIG.  24. — Connections  of  Three-phase  Auto-starter. 

by  the  large  starting  current.  Occasionally  large  starting  fuses  are 
used  in  addition  to  the  running  fuses.  If  this  is  the  case  they  are  con- 
nected in  the  three  taps  leading  from  above  the  running  fuses.  This 
is  usually  not  considered  necessary,  as  a  short-circuit  on  an  alternating- 
current  circuit  is  not  nearly  so  destructive  as  one  on  a  direct-current 
circuit,  on  account  of  the  fact  that  the  reactance  which  is  always  present 
cuts  down  the  current.  A  number  of  taps  are  provided  on  the  trans- 
formers so  that  the  starting  voltage  may  be  varied  to  give  the  starting 
torque  with  as  little  current  as  possible. 

Instead  of  using  a  three-phase  transformer  in  the  way  just  indicated, 
many  manufacturers  use  instead  two  single-phase  transformers  con- 
nected in  open  delta.  This  has  the  advantage  of  allowing  the  same 
transformers  to  be  used  for  a  two-phase  motor.  Fig.  25  shows  the 


46 


THE  INDUCTION  MOTOR 


connections  of  an  auto-starter  of  this  type.  This  method  of  connection 
has  the  advantages  over  that  of  Fig.  24,  that  a  smaller  number  of  con- 
tacts are  required,  and  that  the  heavy  starting  current  of  the  motor 
circuit  does  not  have  to  be  broken. 

RESISTANCE  STARTERS 

In  the  case  of  the  smaller  motors,  say  those  under  30  h.p.,  many 
manufacturers  use  instead  of  the  auto-transformer  as  above  described, 
a  resistance  type  of  starter.  Such  a  device  is  shown  in  Fig.  26.  The 
resistors  are  usually  grouped  in  three  equal  parts  connected  in  the  three 
phases,  and  arranged  to  be  cut  out  in  equal  steps.  This  may  be  done 


FIG.  25. — Connections  of  Three-phase  Auto-starter. 

in  much  the  same  way  as  in  the  case  of  the  direct-current  motor,  by 
dividing  the  resistors  into  many  parts  which  are  successively  cut  out 
of  circuit,  or  the  number  of  steps  may  be  made  much  less  than  is  com- 
mon with  direct-current  motors.  In  fact,  as  far  as  the  motor  and  cir- 
cuit is  concerned,  the  resistance  may  all  be  cut  out  in  one  step,  without 
causing  any  more  disturbance  than  would  be  the  case  if  an  auto-starter 
were  used.  The  connections  must  be  so  made  that  the  fuses  are  out 
of  circuit  during  the  starting  period,  the  same  as  in  the  case  of  the  auto- 
starter. 

There  are  two  other  methods  of  starting  which  deserve  notice. 
In  some  cases  it  is  possible  to  bring  out  taps  from  the  supply  trans- 
formers, giving  a  reduced  voltage  for  starting.  In  this  case  a  three-pole 
double-throw  switch  may  be  used.  Care  should  be  taken  to  see  that 
the  switch  is  of  ample  size  to  carry  and  break  the  excessive  starting 


STARTING  DEVICES  47 

current,  or  bad  burning  of  the  contacts  will  result.  The  principal 
difficulty  in  applying  this  method  is  the  cost  of  running  the  two  or  three 
extra  wires  required,  and  the  fact  that  standard  transformers  are  not 
provided  with  suitable  taps.  In  cases  where  half  voltage  is  sufficient, 
standard  transformers  can  frequently  be  used.  In  this  case,  however, 
since  the  starting  torque  varies  as  the  square  of  the  voltage,  the  starting 
torque  will  be  only  one-quarter  of  the  maximum  the  motor  can  develop, 
and  this  would  usually  be  insufficient. 

Another  method  sometimes  made  use  of  is  to  connect  the  motor 
windings  in  star  for  starting  and  in  delta  for  running.     Since  the 


FIG.  26. — Fairbanks  Morse  Resistance  Type  Starter. 

voltage  over  one  phase  when  the  motor  is  connected  in  star,  is  the 
line  voltage  divided  by  the  square  root  of  three,  the  starting  torque 
will  be  only  one-third  of  the  maximum.  This  would  frequently  be 
insufficient,  and  the  method  lacks  flexibility,  since  this  torque  cannot  be 
changed.  It  is  also  usually  out  of  the  question,  since  most  motors  are 
connected  in  star  rather  than  in  delta,  for  normal  operation.  It  is,  of 
course,  possible  to  develop  a  complete  line  of  motors  with  delta  windings, 
and  use  the  star  delta  method  of  starting  in  all  cases  where  it  will  suffice, 
resorting  to  one  of  the  usual  types  of  starter  when  necessary.  This 
has  been  done  by  the  Crocker-Wheeler  Company,  of  Ampere,  N.  J. 
Another  method  of  starting  used  by  the  Richmond  Electric  Com- 
pany of  Richmond,  Va.,  is  illustrated  in  Fig.  27.  In  addition  to 


48 


THE  INDUCTION  MOTOR 


Line 


the  usual  winding  of  the  motor,  additional  turns  are  wound  on  the  stator 
and  connected  in  series  with  the  stator  winding.  For  starting,  all  of 
the  turns  are  used.  For  running,  connection  is  made  to  the  terminals 
of  the  stator  winding,  and  the  starting  coils  are  left  on  open  circuit. 
With  all  of  the  coils  in  circuit,  the  motor  is  equivalent  to  a  motor  of 
smaller  rating,  and  of  course,  starts  with  a  reduced  current.  The 
motor  might  in  fact  be  operated  indefinitely  with  the  switch  on  the  start- 
ing position,  provided  the  load  were  not  too  great  for  the  reduced  rating. 

The  advantage  of  the  method  is 
that  it  does  away  with  the  need  of  the 
external  starter,  and  permits  of  a  simpler 
starting  switch,  since  the  switch  need  be 
only  three-pole  on  both  sides.  On  the 
other  hand,  in  order  to  accommodate 
the  greater  number  of  turns,  it  is 
necessary  that  the  slots  be  deeper.  This 
in  turn  requires  a  greater  diameter  of  the 
stator  punchings,  and  consequently  a 
larger  motor.  The  leakage  factor  is 
also  slightly  increased,  which  causes  the 
power  factor  to  be  slightly  lower. 

It  should  also  be  mentioned  that  it 
is  frequently  possible  to  effect  a  saving 
by  arranging  to  start  several  motors 
from  the  same  starting  device.  Fig. 
28  shows  the  connections  for  doing 
this.  A  set  of  starting  wires  must 
be  run  to  each  motor  in  addition 
to  the  usual  supply  wires,  and  each 
motor  must  be  provided  with  a  triple-pole,  doublethrow  switch. 
This  switch  is  thrown  to  the  lower  position,  and  the  motor 
started  in  the  usual  way.  The  switch  is  then  thrown  to  the 
upper  position  and  the  starter  returned  tD  the  off-position.  In  some 
cases  when  motors  are  slightly  loaded  for  long  period,  it  might  be 
advisable  to  replace  the  starter  with  a  transformer,  left  constantly  in 
circuit.  Motors  carrying  light  loads  might  then  be  left  running  on 
the  lower  voltage.  The  power-factor  would  thereby  be  very  materially 
raised  and  the  losses  lessened.  Unless  the  motors  are  very  close  together, 
the  expense  of  wiring  in  the  manner  shown  is  apt  to  be  greater  than  the 
cost  of  the  usual  starters.  The  operation  of  starting  is  also  more  com- 


FIG.    27. — Connections  of    the 
Richmond  Induction  Motor. 


STARTING  DEVICES 


49 


plicated,  and  the  apparatus  is  more  apt  to  be  damaged  by  careless 
handling. 

It  should  be  noticed  that  if  an  auto-transformer  is  used,  any  motor 
not  of  greater  rating  than  the  starter  may  be  started  from  it.  Thus 
it  would  be  entirely  proper  to  start  a  10  h.p.  motor  from  a  loo-h.p. 
starter.  This  is  not  true  in  the  case  of  the  resistance  type  of  starter 
unless  the  range  of  resistance  is  very  great.  However,  the  only 
damage  done  would  be  that  if  a  starter  of  a  larger  rating  were  used, 
more  starting  current  than  necessary  would  be  taken  from  the  circuit, 
and  if  the  acceleration  were  too  rapid,  damage  might  be  done  to  the 
belts  or  connected  machinery. 

In  the  case  of  motors  of  5  h.p.  and  less,-  it  is  customary  to  throw 
them  directly  on  the  line.  Frequently,  however,  switches  with  a  double 
set  of  clips  are  used,  so  that  the  fuses  are  cut  out  during  the  operation 


Line 


FIG.  28.— Connections  for  Starting  Several  Motors  with  One  Starter. 


of  starting.  In  this  case,  the  switch  should  be  so  designed  that  it  can- 
not be  left  on  the  starting  position. 

The  question  of  the  relative  advantages  of  the  auto-starter  and  the 
resistance  starter  is  of  importance.  There  has  been  a  disposition  on 
the  part  of  some  central -station  managers  to  object  to  the  resistance 
starter  on  the  ground  that  it  takes  more  current  and  therefore  inter- 
fered more  with  the  regulation  of  the  line  than  the  auto-transformer. 
It  can  be  readily  shown  that  this  is  not  the  case,  and  in  the  writer's  judg- 
ment the  balance  of  advantage  lies  on  the  side  of  the  resistance  starter 
for  small  motors,  and  on  the  side  of  the  auto-starter  for  large  ones. 

There  are  five  important  qualities  that  a  starter  should  have.  These 
the  writer  would  place  in  the  following  order:  Minimum  line  dis- 
turbance, minimum  power  consumption  during  starting,  self-adjusting 
properties,  i.e.,  the  starter  should  automatically  raise  the  voltage  over 


50  THE  INDUCTION  MOTOR 

the  motor  terminals  as  the  motor  speeds  up,  it  should  be  readily  adjust- 
able for  different  starting  torques,  and  it  should  be  low  in  cost. 

In  considering  line  disturbance,  it  should  first  be  noted  that  it  is 
not  the  power  component  of  the  current  that  is  principally  responsible 
for  the  line  drop.  The  drop  is  due  almost  entirely  to  the  wattless 
component  of  the  current.  To  go  into  this  fully  would  require  a  dis- 
cussion of  the  regulation  of  generators,  lines,  and  transformers.  It  is  a 
fact,  however,  that  in  all  of  these  cases,  the  drop  of  pressure  is  due  almost 
entirely  to  the  component  of  current  lagging  90  degrees.  Thus  in  the 
case  of  a  typical  alternator  the  regulation  at  unity  power-factor  might 
be  6  per  cent,  while  at  zero  power-factor,  it  might  be  50  per  cent  or  more. 
The  comparison  would  be  even  more  unfavorable  if  we  took  the  drop  in 
voltage  when  full-load  current  was  thrown  on,  first  at  unity,  and  then 
at  zero  power-factor.  The  same  thing  is  true  of  transformers  and 
transmission  lines.  For  all  practical  purposes  we  may  almost  entirely 
neglect  the  drop  due  to  the  power  component  of  the  current  and  consider 
only  the  wattless  component.  We  shall  now  proceed  to  show  that  this 
wattless  component  is  the  same,  whether  the  resistance  type  or  the 
auto-starter  is  used. 

Let  Ej=line  voltage,  Em=the  voltage  over  the  motor  terminals, 
and  IL  and  Im  the  line  and  motor  currents  respectively,  and  R  and 
Rs  the  resistance  of  the  motor  and  of  the  starting  resistor.  Then  in 
the  case  of  the  auto-starter  we  have 


The  wattless  component  of  the  current  is 


and  if  a  be  the  ratio  of  transformation, 


m         ,     r  W 

a=Jl  and  '---IT' 


then  the  wattless  line  current  is 


STARTING  DEVICES  51 

Similarly  the  power  component  of  the  line  current  is 

a2ELR 

Lp~R2+LW 
and  the  power  is 

P-l,  Ei- 

~  Lp  *~ 

In  the  case  of  the  resistance-type  starter,  let  us  assume  such  a  value 
of  resistance  that  the  voltage  over  the  motor,  the  motor  current  and 
consequently  the  torque  are  the  same  as  before.  The  motor  and  line 
currents  are  now  obviously  the  same,  then 

Em  aEL  EL 


2    \/R2+L2aj2    \ 
The  wattless  component  of  this  current  is 

ELLW 
ILW=  = 


Substituting  in  the  parenthesis  from  the  previous  equation, 

a2ELLa) 
ILW~  R2+L2at  ' 

Similarly  the  power  component  of  the  current  can  be  shown  to  be, 
Lp 


In  the  above  the  wattless  components  of  the  line  current  are  seen 
to  be  the  same  in  the  two  cases,  while  the  power  components  are  in 
the  ratio  of  the  motor  resistance  to  the  resistance  of  the  motor  and  the 
starter.  The  motor  resistance  used  above  is  of  course  the  equivalent 
resistance  of  the  stator  and  the  rotor  combined.  We  thus  see  that  the 
component  of  the  current,  to  which  the  drop  is  principally  due,  is  the 
same  in  the  two  cases.  The  power  component  of  course  produces 
some  drop,  and  this  is  greater  in  the  resistance  type  starter,  but  this  is 
largely,  if  not  entirely  offset  by  the  extra  lagging  current  required  to 


52  THE  INDUCTION  MOTOR 

magnetize  the  transformer,  and  to  produce  the  leakage  flux  of  the 
latter. 

The  writer  has  tested  the  above  deduction  on  a  number  of  motors 
and  starters.  In  no  case  was  the  disturbance  to  the  line  voltage  mate- 
rially different  in  the  two  methods.  If  anything  the  balance  was  slightly 
in  favor  of  the  resistance  starter. 

As  regards  the  second  requirement,  minimum  power  consump- 
tion during  starting,  the  auto-transformer  has  an  undoubted  advantage, 
since  all  the  energy  expended  in  the  starting  resistor  is  wasted.  The 
time  of  starting  is,  however,  usually  so  short  that  the  value  of  this  wasted 
energy  is  not  of  much  moment.  In  cases,  however,  where  the  motor 
is  of  considerable  size  compared  to  the  generator,  the  power  requirement 
during  starting  may  be  so  great  as  to  overload  the  generator  and  engine, 
and  make  the  use  of  the  auto-starter  necessary  or  desirable.  In  present- 
day  practice  the  line  is  usually  drawn  at  about  25  h.p.,  but  there  is  no 
reason  why  with  large  generators,  larger  motors  should  not  be  started 
with  the  resistance  type  starter. 

In  the  third  requirement,  (i.e.,  self  adjustments  of  voltage)  the 
resistance  starter  is  clearly  superior.  This  is  due  to  the  fact  that  as 
the  motor  speeds  up,  the  current  decreases,  due  to  the  counter  e.m.f. 
increasing,  and  consequently  the  voltage  applied  to  the  motor  terminals 
increases  automatically.  This  means  that  when  the  motor  is  finally 
thrown  directly  on  the  line,  the  sudden  change  in  voltage  will  not  be 
so  great  and  consequently  the  supply  voltage  will  not  be  disturbed  to 
so  great  an  extent  as  with  the  auto-starter.  If  the  starter  is  supplied 
with  a  number  of  intermediate  steps,  the  disturbance  is  of  course  still 
less. 

In  regard  to  the  feature  of  ready  adjustability,  the  resistance  starter, 
if  it  is  provided  with  a  number  of  intermediate  steps,  is  evidently  supe- 
rior, since  it  is  inherently  self-adjusting.  It  should  be  set  to  give  on  the 
first  point  about  70  per  cent  of  full-load  torque.  If  more  than  this  is 
needed,  the  attendant  at  once  secures  it  by  moving  the  starting  arm 
to  the  required  notch.  Thus  no  more  current  is  taken  than  is  absolutely 
required.  This  feature  is  particularly  valuable  in  cases  where  the 
starting  torque  is  apt  to  vary  widely  at  different  times. 

It  is  evident  that  as  regards  the  question  of  cost,  the  resistance 
starter  can  be  manufactured  at  a  lower  cost  than  the  auto-starter. 
This  fact,  together  with  some  of  the  points  brought  out,  will,  it  is 
believed,  insure  a  large  use  for  it,  at  least  in  the  smaller  sizes. 


STARTING  DEVICES  53 


CARBON  BLOCK  STARTERS 

Since  the  foregoing  was  written,  a  new  type  of  resistance  starter 
for  induction  motors  has  been  put  on  the  market  by  the  American 
Electric  Fuse  Co.,  of  Muskegon,  Mich.  Starters  embodying  the  same 
principle  are  also  built  for  use  with  direct-current  motors.  The 
essential  feature  of  this  device  is  the  employment  of  the  imperfect 
contact  between  carbon  blocks  to  produce  the  resistance.  It  is  well 
known  that  if  two  carbon  blocks  are  pressed  together  with  a  small 
force  the  resistance  to  a  current  will  be  large,  and  if  this  force  is  increased 
the  resistance  will  be  very  much  reduced.  It  is  possible  in  this  way  to 
obtain  a  variation  of  resistance  of  as  much  as  200  to  i,  without  employ- 
ing such  small  pressures  as  to  render  the  contact  uncertain.  The 
same  principle  has  been  utilized  for  a  number  of  years  in  various  devices, 
notably  hi  the  telephone  transmitter. 

Considered  from  a  theoretical  standpoint,  this  construction  seems 
to  offer  a  number  of  advantages.  Of  these,  perhaps  the  most  apparent 
is  the  fact  that  all  of  the  resistor  is  in  use  all  of  the  time.  Each  resistor 
unit  has  a  definite  rating  in  watts,  the  value  of  this  rating  depending, 
of  course,  on  the  nature  of  the  service,  i.e.,  whether  intermittent  or 
continuous.  It  is  entirely  independent  of  the  value  of  the  current  or 
of  the  e.m.f.,  and  depends  only  on  their  product.  Thus  one  unit  can 
be  used  for  a  variety  of  different  currents  and  e.m.fs.,  providing  only 
that  the  rated  power  in  watts  is  not  exceeded.  This  is  not  true  of  the 
more  usual  form  of  rheostats,  since  if  the  current  is  increased  by  cutting 
out  a  portion  of  the  resistor  units,  the  watt  rating  of  the  apparatus  is 
reduced,  because  the  units  cut  out  are  then  not  available  for  radiating 
the  energy  lost  in  the  resistance. 

From  the  standpoint  of  the  user,  this  means  that  it  is  possible  to 
start  the  motor  whether  a.c.  or  d.c.  with  the  minimum  amount  of  cur- 
rent, and  consequently  with  the  minimum  chance  of  injury  to  the 
connected  apparatus.  From  the  standpoint  of  the  electricity  supply 
company,  it  means  that  the  motor  demands  during  the  starting  period, 
only  the  minimum  amount  of  current  and  power  that  will  possibly  start 
it.  We  have  shown  in  the  case  of  the  polyphase  induction  motor, 
that  the  amount  of  wattless  current  is  the  same  whether  a  resistance 
type  or  a  transformer  type  starter  is  used.  Since  in  the  case  of  a  starter 
of  the  kind  just  described,  only  the  minimum  amount  of  starting 
current  is  used,  it  follows  that  with  a  carbon-block  starter  the  amount 
of  a  wattless  current  will  in  general  be  less  than  in  the  case  of  the  trans- 


54 


THE  INDUCTION  MOTOR 


former  starter,  and  consequently  the  line  disturbance  will  in  general 
be  less.  In  the  limiting  case,  when  the  transformer  starter  is  so  set 
that  it  can  just  start  the  load,  the  wattless  current  and  the  line  dis- 
turbance will  be  the  same. 

The  other  advantages  mentioned  in  connection  with  the  resistance- 
type  starter  apply  in  this  case  also.  A  perspective  view  of  one  of  these 
starters  is  shown  in  Fig.  29.  It  will  be  noted  that  there  are  six  con- 
nectors at  the  top  of  the  starter.  The  three  extra  terminals  are  for  the 


FIG.  29. — Connections  of  American  Electric  Fuse  Company's  Starter. 

insertion  of  the  starting  fuses  or  to  allow  the  lines  to  be  connected  back 
of  the  fuses  for  starting.  During  the  time  of  starting,  the  running 
fuses  are  short-circuited  by  the  three  small  levers  shown.  Fig.  30 
shows  the  curves  of  time  and  power  during  the  starting  of  a  i5-h.p. 
44o-volt  three-phase  induction  motor  with  a  carbon-block  rheostat 
and  with  an  auto-starter.  The  motor  was  started  under  a  heavy  load 
consisting  of  a  line  shaft  and  counter-shafting.  The  load  was  of  course 
the  same  in  the  two  cases.  The  auto-starter  was  one  that  would  show  up 
to  the  best  advantage,  since  it  was  of  the  three-point  type,  thus  afford- 
ing smoother  acceleration  than  would  have  been  the  case  if  the  ordinary 
type  of  two-point  starter  has  been  used.  The  high  peaks  of  power  as 


STARTING  DEVICES 


55 


the  auto-starter  was  thrown  successively  to  the  three  starting  positions 
are  very  apparent.  In  the  case  of  the  carbon-block  starter,  on  the  other 


FIG.  30. — Starting  Currents  of  an  Induction  Motor  with  American  Electric  Fuse 
Co's.  Carbon  Block  Starter  and  with  Three-point  Auto-starter. 


Flexible  Ju 


VJ  in.  Graphite 


nsulating  Sleeve 
Steel  Tube 


in.  Graphite  Disks 


M  in.  Graphite  Dis 


Connection  St 
Insulating  W 


in.  Graphite  Disks 
Graphite  Plug 
Bottom  Cap 

Mica  Washer 


Iron  Washer 
FIG.  31. — Section  of  American  Electric  Fuse  Go's.  Resistance  Unit. 

hand,  the  power  taken  can  be  kept  nearly  uniform,  and  the  sudden 
jumps  in  power  and  consequently  in  the  voltage  can  be  avoided.     In 


56  THE  INDUCTION  MOTOR 

Fig.  31  is  shown  a  section  of  one  of  the  resistance  tubes.  From  this 
the  construction  can  be  clearly  seen.  As  will  readily  be  surmised,  the 
principal  difficulty  in  the  development  of  this  piece  of  apparatus,  was 
the  production  of  a  suitable  lining  for  the  iron  tube  used  to  contain 
the  disks,  since  it  is  necessary  that  these  be  insulated  from  the  rest  of 
the  starter.  Given  a  suitable  lining,  the  apparatus  is  almost  indestruct- 
ible. The  writer  understands  that  this  lining  has  been  brought  to  a 
high  state  of  perfection. 

PROTECTION  AGAINST  OVERLOADS  IN  INDUCTION  MOTORS 

The  provision  of  proper  protection  (for  the  squirrel-cage  induction 
motor)  is  a  more  difficult  problem  than  in  the  case  of  the  direct-current 
machine.  This  is  on  account  of  the  fact  that  the  squirrel-cage  induction 
motor  requires  a  much  greater  current  during  the  starting  period  than 
it  does  while  in  normal  operation.  The  starting  current  may  amount 
to  as  much  as  five  times  the  running  current  and  a  fuse  which  would 
afford  protection  during  starting  would  be  of  little  or  no  use  during 
ordinary  operation.  Owing  to  the  great  difficulty  of  arriving  at  a 
satisfactory  solution,  the  National  Board  of  Fire  Underwriters  have 
apparently  hesitated  to  make  rigid  rules  which  might  hinder  the  develop- 
of  the  industry.  As  a  consequence,  the  regulations  are  not  nearly  so 
rigid  as  is  the  case  with  direct-current  motors.  Thus  the  practice  of 
starting  with  no  fuses  in  circuit  is  at  least  tolerated,  and  a  size  of  wire 
supplying  the  motor  corresponding  in  current-carrying  capacity  to  the 
starting  current  of  the  motor  is  rarely  required. 

The  provision  requiring  a  no-voltage  release  on  all  direct-current 
motor  starters  is  not  enforced  in  the  case  of  the  alternating-current 
motors.  The  object  of  the  release  is  that  if  the  power  is  temporarily 
removed  from  the  line  the  starter  may  return  to  the  off-position.  When 
the  power  is  again  restored,  the  motor  will  remain  disconnected  until 
properly  started  by  an  attendant.  The  induction  motor  would, however, 
not  be  damaged  by  having  the  power  thrown  on  under  such  circum- 
stances, and  about  all  the  harm  that  would  be  done  would  be  the  blow- 
ing of  the  fuses.  Hence,  it  is  doubtful  if  such  a  provision  would  be 
warranted. 

The  use  of  fuses  in  connection  with  any  motor  is  open  to  some 
serious  objections,  and  the  principal  point  in  their  favor  is  their  low 
first  cost.  If  used  under  circumstances  where  they  are  frequently 
blown,  the  cost  of  renewals  may  be  a  very  considerable  item,  and  may 
render  the  use  of  a  circuit-breaker  preferable. 


STARTING  DEVICES  57 

When  used  with  induction  motors,  there  is  an  objection  to  the  fuse 
which  does  not  apply  in  the  case  of  direct-current  machines.  In  the 
latter  case,  if  one  of  the  fuses  is  burned  out  the  motor  will  not  start, 
or  if  in  operation, it  will  at  once  stop.  In  the  case  of  the  polyphase 
induction  motor,  since  the  motor  does  not  take  its  starting  current 
through  the  running  fuses,  one  (or  more)  of  the  latter  may  be  burned 
out  and  the  motor  will  start  just  as  though  nothing  were  the  matter. 
After  the  motor  has  attained  nearly  full  speed,  the  starter  handle  will  be 
thrown  to  the  running  position,  and  if  two  of  the  fuses  of  a  three-phase 
motor,  or  the  two  fuses  of  one  phase  of  a  two-phase  motor  are  ntact, 
the  motor  will  continue  to  operate  as  a  single-phase  machine.  This 
condition  may  readily  escape  notice,  since  the  motor  appears  to  operate 
normally.  If  the  fuses  are  inspected  the  indicator  used  with  enclosed 
fuses  may  have  failed  to  operate  and  the  fuse  will  consequently  be 
apparently  all  right.  Even  though  a  test  is  made,  some  electricians 
will  test  with  a  voltmeter  over  each  of  the  fuses  while  the  motor  is  in 
operation.  There  will,  of  course,  be  little  or  no  voltage  over  the  defective 
fuse,  and  this  test  will  show  everything  apparently  all  right.  It  may 
appear  that  undue  emphasis  is  laid  upon  a  remote  possibility,  but 
any  engineer  who  has  had  much  to  do  with  induction  motor  troubles  can 
testify  that  cases  of  this  nature  very  frequently  occur. 

When  this  condition  is  present,  the  winding  of  the  motor  still  in 
circuit  is  apt  to  be  overloaded  and  damaged.  It  might  appear  that  on 
account  of  the  overloading  of  the  remaining  phase,  the  fuses  on  this 
phase  would  open  and  relieve  the  motor.  This,  of  course,  often  happens, 
but  in  many  other  cases,  it  is  found  that  advantage  has  been  taken  of 
the  sturdy  character  of  the  induction  motor  and  heavier  fuses  than  should 
normally  be  used  have  been  substituted.  This  may  have  been  done 
on  account  of  the  trouble  frequently  experienced  by  fuses  blowing  when 
the  starter  is  thrown  from  the  starting  to  the  running  position. 

By  using  a  circuit-breaker,  the  possibility  of  this  occurrence  can 
be  almost  entirely  prevented.  To  take  care  of  the  starting  period, 
taps  can  be  brought  out  of  the  trip  coils,  so  that  only  a  portion  of  the 
turns  are  in  circuit  while  the  motor  is  being  started.  This  can  be  done 
at  a  small  expense,  and  affords  at  least  partial  protection  during  start- 
ing. The  rush  of  current  at  the  instant  when  the  starter  is  thrown 
from  the  starting  to  the  running  position,  is  more  difficult  to  deal  with. 
Some  provision  may  be  made  by  which  the  attendant  may  temporarily 
short-circuit  some  of  the  turns  of  the  trip  coils,  the  device  being  so 
constructed  that  it  will  immediately  throw  these  turns  in  circuit  as 


58  THE  INDUCTION  MOTOR 


FIG.  32.— I.T.I.E.  Time  Limit,  no  Voltage  and  Overload,  Circuit  Breaker,  Built 
by  the  Cutter  Co. 


FIG.  33.— Westinghouse  Auto-starter  Switch. 


STARTING  DEVICES  59 

soon  as  the  attendant  leaves  the  starter.  Several  other  solutions 
along  the  same  line  have  been  proposed. 

One  of  the  devices  which  seems  most  promising  is  the  use  of  a  time- 
limit  relay  in  connection  with  the  circuit-breaker.  With  the  addition 
of  this  device,  the  breaker  will  not  at  once  open  on  the  application  of 
the  overload,  but  it  will  be  necessary  that  the  condition  of  overload 
continue  a  definite  time,  before  the  breaker  will  operate.  This  is  of 
some  advantage  in  the  normal  operation  of  the  motor,  since  it  is  not  nec- 
essary to  set  the  breaker  for  a  large  current,  in  order  to  take  care  of 
harmless  temporary  overloads.  It  is,  however,  of  the  greatest  benefit 
in  taking  care  of  the  starting  period  in  a  satisfactory  manner. 

A  breaker  of  this  type  manufactured  by  The  Cutter  Company,  and 
adapted  for  use  on  circuits  of  550  volts  and  under  is  shown  in  Fig.  32. 
A  Westinghouse  switch  adapted  for  use  on  22oo-volt  circuits  is  shown 
in  Fig.  33.  In  this  case,  the  switch  is  of  the  double-throw  variety 
with  six  contacts  on  one  side  and  three  on  the  other  in  the  case  of  a 
three-phase  outfit.  The  same  switch  will  then  serve  both  as  a  start- 
ing switch  and  as  a  circuit-breaker  to  protect  the  motor  in  regular 
operation.  The  use  of  such  combined  breakers  and  starting  switches 
on  22oo-volt  circuits  is  quite  common,  but  on  account  of  the  cost  and 
the  recent  introduction  -of  the  apparatus,  the  fuse  protection  is  more 
usual  on  low-voltage  circuits. 

With  high-voltage  outfits,  the  use  of  oil  as  a  medium  to  assist  in 
breaking  the  arc  is  almost  universal.  Most  of  the  makers  have  followed 
the  same  practice  in  the  case  of  the  low-voltage  breakers.  This  has 
probably  been  largely  due  to  the  fact  that  the  motor  itself  having  no 
sliding  connections  is  free  from  the  possibility  of  sparking  and  the 
attendant  fire  risks.  This  is  of  importance  in  certain  industries,  and 
naturally  leads  to  the  wish  to  obtain  the  same  desirable  property  in  the 
protective  device.  In  places  where  an  open  arc  can  do  no  harm  in 
the  event  of  the  breaker  opening,  there  is  apparently  no  reason  why  the 
breaker  with  the  break  in  air  should  not  be  equally  reliable.  Certain 
incidental  advantages  such  as  the  removal  of  the  possibility  of  the 
oil  leaking  out  without  being  noticed,  and  being  absent  when  most 
wanted,  are  secured  by  the  air  break.  The  fact  that  it  is  at  once  apparent 
whether  or  not  the  breaker  is  open,  is  also  of  some  advantage. 

The  above  remarks  apply  in  general  to  the  squirrel-cage  type  of 
motor.  The  wound-rotor  machines  can  be  handled  along  the  same 
lines  found  desirable  with  direct-current  machines,  since  the  starting 
current  is  not  necessarily  much  greater  than  the  running  current. 


CHAPTER  V 
THE  INDUCTION  GENERATOR 

ANY  induction  motor  may  be  operated  as  a  generator.  In  Fig. 
34  is  shown  the  circle  diagram  of  an  induction  motor,  but  extended  to 
include  the  action  of  the  machine  as  a  generator.  The  line  OB  repre- 
sents in  magnitude  and  phase  the  current  taken  by  the  machine  when 


Applied  E.M.F. 


Generated  E.M.F. 
FIG.  34. — Circle  Diagram  of  Induction  Motor  or  Induction  Generator. 

acting  as  a  motor  under  approximately  full  load.  The  line  OA 
represents  the  current  at  synchronism.  If  now  the  motor  be  forced 
to  run  at  a  speed  in  excess  of  synchronism,  the  current  vector  will 
be  represented  by  some  such  line  as  OC.  A  component  of  the  current 
is  now  in  the  same  phase  as  the  generated  or  counter  e.m.f.  or  the 
machine  is  acting  as  a  generator. 

That  this  will  be  so  is  also  readily  seen  from  a  consideration  of 
Fig.  7,  page  6.  It  will  be  apparent  that  if  the  rotor  be  forced  to 
revolve  at  a  greater  speed  than  synchronism,  the  rotor  conductors  will 
cut  the  flux  as  before,  but  in  the  opposite  direction.  The  e.m.f.  gen- 

60 


THE  INDUCTION  GENERATOR  61 

crated  and  consequently  the  current  will  be  reversed  in  the  rotor,  and 
this  will  require  a  power  component  of  current  in  the  stator  in  the 
reverse  direction  to  offset  it.  Since  the  current  in  the  rotor  is  reversed, 
the  torque  is  also  reversed,  or  the  machine  acts  as  a  generator. 

Several  things  will  be  at  once  apparent  from  an  inspection  of  the 
diagram.  The  maximum  output  as  a  generator  will  not  be  so  large  as 
the  maximum  input  as  a  motor.  If  this  output  is  exceeded,  and  the 
torque  applied  to  the  motor  is  maintained,  the  machine  will  speed  up 
indefinitely.  Of  course  in  practice  this  would  not  be  the  case,  as  the 
governor  of  the  prime  mover  would  act  to  limit  the  speed.  It  is  also 
evident  from  the  diagram  that  the  maximum  power-factor  will  be  slightly 
less  as  a  generator  than  as  a  motor. 

An  induction  generator  of  this  type  is  not  self -exciting.  Both  its 
voltage  and  its  frequency  are  fixed  by  the  voltage  and  frequency  of  the 
line  to  which  it  is  connected.  The  power  it  delivers  is  determined  by 
the  amount  by  which  it  exceeds  synchronous  speed.  If  it  were  desired 
to  operate  a  station  using  only  induction  generators,  it  would  be  neces- 
sary to  provide  a  sufficient  capacity  in  synchronous  machines  to  supply 
the  magnetizing  current  of  the  induction  generators.  The  combined 
capacity  of  the  synchronous  machine  would  therefore  need  to  be  approx- 
imately 25  per  cent  of  the  capacity  of  the  induction  machines,  provided 
the  power-factor  of  the  connected  load  were  unity.  The  exciters 
could  take  the  form  of  synchronous  motors,  running  idle  on  the  line, 
or  they  might  be  of  somewhat  greater  rating,  and  be  used  to  supply  a 
certain  amount  of  power  to  the  station  busbars,  or  be  used  as  motors, 
in  addition  to  furnishing  the  magnetizing  current  of  the  induction 
generators. 

It  will  be  seen  that  the  current  supplied  by  the  induction  generator 
is  leading,  considering  the  machine  as  a  generator.  That  this  is  so 
will  be  apparent  from  Fig.  34.  The  line  OD  represents  the  phase  of 
the  applied  e.m.f.  when  the  machine  is  running  as  a  motor.  The  back 
e.m.f .  of  the  machine  is  represented  by  the  line  OE,  and  when  the  machine 
is  acting  as  a  generator,  this  becomes  the  terminal  e.m.f.  The  direc- 
tion of  rotation  of  the  vectors  was  taken  as  counter-clockwise,  and 
consequently  the  current  now  leads  the  e.m.f. 

A  possible  arrangement  of  the  apparatus  is  shown  in  Fig.  35.  We 
have  here  an  induction  generator  electrically  connected  to  a  synchronous 
machine,  and  the  combination  connected  to  the  line.  The  two  machines 
must  not  be  mechanically  connected,  but  must  be  free  to  rotate  at  the 
proper  speed  to  give  the  required  slip  to  the  induction  machine. 


62  THE  INDUCTION  MOTOR 

Suppose  for  the  minute  that  there  is  no  load  on  the  line,  and  that  the 
synchronous  machine  is  not  mechanically  driven,  but  acts  only  as  a 
synchronous  motor.  In  order  to  supply  its  losses,  the  synchronous 
machine  will  require  a  small  component  of  power  current,  and  the 
induction  machine  will  require  a  much  larger  component  of  wattless 
current  to  magnetize  it.  This  current  is  leading  the  e.m.f.  of  the 
induction  generator.  Consequently  the  synchronous  machine  must 
supply  a  lagging  current  considered  as  a  generator,  or  take  a  leading 
current  considered  as  a  motor.  Its  current  is  thus  almost  entirely 
wattless,  and  its  field  must  be  strengthened  beyond  the  point  that 
would  be  necessary  if  it  were  operating  at  unity  power-factor.  We  may, 
in  fact,  consider  that  in  a  sense  the  field  excitation  of  the  synchronous 

To  Load  


FIG.  35. — Connections  of  Exciter  and  Induction  Generator. 

machine  must  be  sufficient  to  force  the  required  flux  across  the  air- 
gaps  of  both  machines. 

If  now  a  load  be  connected  to  the  generator,  the  conditions  may 
be  materially  changed.  The  character  of  the  connected  load  deter- 
mines the  nature  of  the  current  supplied.  If  the  load,  for  example, 
consists  of  incandescent  lamps  only,  the  power  factor  will  be  practically, 
unity.  If  it  consists  of  induction  motors,  the  current  will  be  lagging, 
or  if,  on  the  other  hand,  the  load  consists  of  synchronous  motors  with 
overexcited  fields  or  of  a  non-inductive  load  at  the  end  of  a  long  trans- 
mission line,  the  current  may  be  leading.  The  induction  generator, 
it  will  be  noted  from  Fig.  34,  can  furnish  current  at  only  one  power- 
factor  for  each  value  of  the  load.  This  current  consists  of  a  component 
in  phase  with  the  e.m.f.,  and  a  component  leading  it  by  90  degrees. 
This  leading  component  is  nearly  constant,  but  increases  somewhat 
as  the  load  increases.  If  the  connected  load  takes  a  current  having  a 
wattless  component  differing  from  that  which  the  generator  can  supply, 
the  excess  or  deficiency  of  wattless  current  must  be  furnished  by  the 


THE  INDUCTION  GENERATOR  63 

synchronous  machine.  Thus  if  the  current  supplied  by  the  generator 
be  represented  by  OC,  it  consists  of  a  power  component  OF  and  a  watt- 
less leading  component  FC.  The  power-factor  of  the  generator  cur- 
rent is  OF  divided  by  OC.  If  the  load  demands  a  current  of  this  same 
power  factor,  the  synchronous  machine  will  supply  no  current.  This 
neglects  of  course  the  small  losses  of  this  m?.chine. 

If  on  the  other  hand  the  circuit  demands  a  lagging  current,  the 
synchronous  machine  must  supply  not  only  the  leading  current  required 
by  the  induction  machine,  but  also  enough  leading  current  to  bring  the 
power-factor  of  the  whole  output  to  that  value  of  the  power-factor 
which  the  generator  is  capable  of  furnishing  at  the  load  considered.  In 
general  this  current  will  be  considerable,  since  in  the  great  majority  of 
cases  the  current  required  is  lagging,  and  in  this  case  the  synchronous 
machine  must  furnish  not  only  all  the  lagging  current  for  the  load,  but  in 
addition  the  current  required  to  excite  the  induction  machine. 

The  question  will  arise  in  the  reader's  mind,  since  a  leading  current 
is  required  for  excitation,  why  is  it  not  practicable  to -make  the  induc- 
tion generator  a  self-excited  machine  by  providing  a  condenser  to 
consume  the  leading  current  required  ?  This  could  be  done  without  an 
adjustable  condenser,  although  the  cost  would  be  very  great,  providing 
the  power-factor  of  the  load  to  be  supplied  did  not  change.  What 
would  be  required  in  a  practical  case,  would  in  fact,  be  a  combina- 
tion of  a  condenser  and  an  inductance,  both  capable  of  being 
adjusted  to  any  value  required,  and  of  sufficient  size  so  that  the  power- 
factor  of  the  load,  combined  with  the  regulating  inductance  and  con- 
denser, could  always  be  brought  to  that  value  of  power-factor  which 
the  generator  was  capable  of  supplying  under  the  given  load.  It  will 
be  seen  at  once  that  if  the  load  were  of  such  a  character  that  the  power- 
factor  were  at  all  variable,  the  size  and  cost  of  such  a  regulating  device 
would  be  prohibitive. 

Such  a  combination  would  also  be  somewhat  unstable  in  voltage, 
since  the  only  factor  which  would  tend  to  hold  the  voltage  constant  is 
the  fact  that  the  power-factor  of  the  induction  generator  changes  slightly 
with  a  change  of  voltage.  This  is  true  since  the  ratio  of  the  magnetizing 
current  to  the  locked  current  changes  somewhat  with  change  of  voltage 
on  account  of  the  saturation  of  the  iron.  This  increases  the  magnetizing 
current  for  greater  voltages  in  a  greater  proportion  than  it  increases 
the  locked  current.  Since  the  maximum  power-factor  is  equal  to 

— - — >  and  since  a =——  it  will  be  seen  at  once  that  for  each  value  of  the 
1  +  20  AG 


64  THE  INDUCTION  MOTOR 

power-factor  of  the  load,  the  generator  would  tend  to  assume  a  certain 
voltage.  This  voltage  would,  however,  not  be  as  definite  as  might  be 
desirable,  since  the  iron  of  the  induction  motor  is  not  in  general  worked 
at  such  a  density  as  to  reach  the  knee  of  the  magnetization  curve.  It 
would  be  more  definite  on  25  cycles  than  on  60  cycles,  since  in  the  former 
case  higher  densities  are  employed  than  in  the  latter. 

The  arrangement  using  a  synchronous  machine  for  excitation  is  like- 
wise open  to  serious  objection,  on  account  of  the  cost  of  the  two  machines, 
and  on  account  of  the  poor  regulation  of  the  combination.  It  is  evident 
that  by  increasing  the  size  of  the  synchronous  machine  somewhat,  pro- 
viding it  with  a  prime  mover,  and  using  it  to  supply  a  part  of  the  power 
required,  that  the  cost  per  k.w.  of  output  will  be  reduced.  For  instance 
if  the  synchronous  machine  were  furnishing  100  k.v.-amp.  of  wattless 
current,  by  increasing  its  size  to  141  k.v.-amp.  it  would  be  capable  of  fur- 
nishing the  same  wattlers  current  as  before,  and  at  the  same  time  it  could 
carry  a  power  component  of  current  corresponding  to  an  output  of  100 
k.w.  This  follows  since  the  two  currents  are  90  degrees  apart  in  phase 
and  their  resultant  is  therefore  equal  to  V  2  =  1.41  times  as  great  as 
either.  We  should  therefore  obtain  a  power  rating  at  100  k.w.  at  the 
cost  of  an  increase  in  size  corresponding  to  41  k.w.  It  is  doubtful, 
however,  whether  the  conditions  would  ever  be  such  as  to  justify  an 
installation  of  this  kind. 

In  the  case  of  a  plant  already  installed,  to  which  it  is  desired  to  add 
another  machine,  the  case  against  the  induction  generator  is  not  so 
strong.  Consider  for  example  a  large  plant  delivering  its  energy  through 
a  long  transmission  line,  and  suppose  it  is  desired  for  some  reason  to  feed 
into  it  from  a  generator  driven  by  a  gas  engine.  Owing  to  high 
frequency  or  the  great  resistance  and  reactance  of  the  lines  between  the 
main  generators  and  the  gas  engine  set,  hunting  of  the  latter  unit  may  be 
feared.  The  use  of  an  induction  generator  would  largely  remove  this 
danger.  In  such  a  system,  moreover,  it  is  entirely  possible  that  the  cur- 
rent taken  by  the  line  and  load  may  normally  be  leading,  on  account  of 
the  capacity  of  the  line.  In  this  case  the  induction  generator,  since  it 
gives  a  leading  current,  will  supply  at  least  a  part  of  the  leading  current 
required,  and  hence  will  relieve  the  main  generators.  Some  difficulty 
will  be  experienced  if  an  attempt  is  made  to  govern  the  gas  engine,  since 
it  would  be  necessary  to  use  a  governor  such  that  the  speed  would 
increase  with  the  load.  Probably  the  most  satisfactory  method  of 
operating  would  be  to  run  the  gas  engine  at  constant  load,  merely  pro- 
viding a  hand  throttle  and  a  speed  limiting  device  to  prevent  the  running 


THE  INDUCTION  GENERATOR  65 

away  of  the  engine  in  case  the  breakers  should  open  and  remove  the  load 
from  the  generator. 

A  similar  problem  is  sometimes  encountered  in  the  case  of  small 
water  powers.  It  frequently  happens  that  it  would  be  feasible  and  cheap 
to  develop  such  powers,  but  the  expense  of  an  attendant  would  more 
than  offset  the  value  of  the  power  generated.  In  such  a  case  we  could 
make  use  of  an  outfit  consisting  of  an  induction  generator,  either  direct 
connected  or  belted  to  a  water  wheel,  and  provided  with  suitable  means 
for  preventing  the  speed  rising  much  above  synchronism  in  case  the 
power  goes  off  the  line  or  the  circuit-breakers  open.  Such  a  plant 
should  give  satisfactory  service  with  only  a  daily  inspection.  It  is  of 
course  necessary  that  the  rating  of  the  induction  generator  should  be 
small  in  comparison  with  that  of  the  synchronous  generators,  and  if  it  is 
not  provided  with  a  governor,  it  is  essential  that  the  power  required  on 
the  line  should  never  be  less  than  the  rating  of  the  induction  machine. 

Another  and  one  of  the  most  promising  of  the  uses  to  which  the  induc- 
tion generator  has  been  put,  is  in  connection  with  turbine-driven  direct- 
current  generators.  It  is  well  known  that  the  construction  of  such  a 
generator  presents  serious  difficulties.  These  arise  principally  from  two 
causes,  the  difficulty  of  constructing  an  armature  and  commutator  which 
will  safely  withstand  the  great  strains  due  to  centrifugal  force,  and  the 
difficulty  of  commutating  the  current.  It  is  impossible  in  large  machines 
to  keep  the  voltage  between  commutator  bars  down  to  a  reasonable  value, 
since  at  least  one-half  a  turn  must  be  included  between  bars  (and  to  do 
even  this  requires  a  rather  awkward  construction  with  taps  extending 
from  the  commutator  to  the  rear  of  the  coil),  and  the  voltage  generated 
in  even  one-half  turn  is  frequently  far  in  excess  of  a  reasonable  value. 

It  is  obvious  that  a  substitute  may  be  found  by  employing  a  synchron- 
ous alternator,  and  rectifying  the  current  by  means  of  a  rotary  con- 
verter. This  scheme,  while  perfectly  workable,  has  certain  disadvantages. 
Since  both  machines  are  of  the  synchronous  variety,  trouble  may 
be  experienced  on  account  of  hunting,  although  no  serious  difficulty 
is  apt  to  develop.  A  certain  amount  of  skill  and  care  are  necessary  in 
synchronizing  the  machines,  and  there  is  always  the  possibility  of  damage 
being  done,  due  to  lack  of  care  in  this  respect.  The  most  serious  diffi- 
culty, however,  is  encountered  in  the  regulation  of  the  voltage.  In  order 
that  there  should  not  be  large  wattless  currents  circulating  between  the 
generator  and  converter,  it  is  essential  that  the  field  excitation  on  the 
two  machines  should  be  such  that  they  would  give  approximately  the 
same  voltage  on  open  circuit.  In  order  to  preserve  this  equality,  it 


66  THE  INDUCTION  MOTOR 

•would  probably  be  advisable  to  provide  some  form  of  mechanical  con- 
nection between  the  two  field  rheostats.  Moreover,  it  would  be  imprac- 
ticable to  compound  the  converter  unless  reactance  coils  were  used 
between  the  converter  and  the  generator.  If  this  were  done  the  con- 
verter would  operate  at  unity  power-factor  at  only  one  load. 

Most  of  these  difficulties  are  removed  by  using  an  induction  generator 
in  place  of  the  synchronous  machine.  To  start  the  set  it  is  merely  neces- 
sary to  run  the  induction  generator  at  any  convenient  speed,  run  the 
converter  by  some  outside  means  up  to  approximately  the  corresponding 
frequency  connect  the  two  together,  and  excite  the  field  of  the  converter. 
In  fact  it  would  probably  be  preferable  to  consider  the  converter  as  part 
of  the  generator  and  provide  permanent  electrical  connections,  that  is, 
have  no  switches  between  the  generator  and  the  converter.  It  would  also 
be  preferable  to  connect  the  generator  for  a  large  number  of  phases,  say 
six  or  more.  This  will  greatly  reduce  the  copper  loss  of  the  converter,  or 
conversely  the  rating  of  the  converter  can  be  greatly  increased  for  a  given 
size  of  armature,  by  simply  providing  a  large  enough  commutator  to  take 
care  of  the  increased  output. 

In  the  last  few  years  cases  similar  to  the  above  have  arisen,  in  con- 
nection with  certain  central  stations  in  which  the  generating  units  were 
driven  by  reciprocating  engines.  A  reciprocating  engine  is  excellently 
adapted  to  utilize  steam  efficiently  in  the  range  of  pressures  between 
the  usual  boiler  pressure  and  the  pressure  of  the  atmosphere.  The 
turbine,  on  the  other  hand,  is  better  adapted  to  handle  the  steam  in 
the  range  between  atmosphere  and  zero  pressure.  Moreover,  in  many 
cases  it  was  found  that  it  was  possible  to  double  the  rating  of  the  station 
by  installing  a  turbine  between  each  pair  of  reciprocating  engines,  and 
still  leave  sufficient  room  for  the  operation  of  the  plant. 

If  the  plant  is  to  operate  in  this  way  with  each  of  the  reciprocating 
engines  exhausting  into  one  of  the  turbines,  it  is  essential  that  one 
engine  and  its  corresponding  turbine  be  regarded  as  a  unit.  The  tur- 
bine is  operated  without  a  governor,  taking  all  of  the  steam  exhausted 
by  the  engine. 

If  the  generator  operated  by  the  turbine  were  a  synchronous  machine, 
it  will  be  seen  at  once  that  the  combination  might  give  trouble  in  various 
ways.  The  use  of  an  induction  generator  in  such  a  combination  offers 
numerous  advantages.  A  notable  example  of  such  an  installation  is 
afforded  in  the  case  of  the  station  of  the  Interborough  Rapid  Transit  Co. 
of  New  York  City.  The  power  house  contains  (9)  units  each  of  (7500) 
k.w.,  and  as  noted  each  unit  consists  of  a  reciprocating  engine  exhausting 


THE  INDUCTION  GENERATOR 


67 


into  a  low-pressure  turbine.  The  generators  used  on  the  reciprocating 
engines  are  of  the  usual  synchronous  type,  but  those  used  in  connection 
with  the  turbines  are  induction  generators.  The  induction  generators 
are  connected  to  the  synchronous  machines  by  simple  knife  switches, 
and  these  switches  are  not  intended  to  be  used  in  normal  operation. 
To  start  a  set,  the  field  of  the  synchronous  generator  is  first  excited,  and 
steam  is  then  admitted  to  the  engine.  The  induction  machine  starts  as 
an  induction  motor  until  steam  reaches  it,  when  it  automatically  begins 
operation  as  a  generator. 


PHASE  CONVERTER 

It  will  be  readily  seen  that  an  induction  motor  might  be  wound  for 
two  or  more  different  phases.  Thus  a  machine  could  be  wound  with 
twelve  sets  of  coils  per  pair  of  poles  and  be  connected  as  in  Fig.  36.  If 

connection  were  made  to  points 
AB  and  C,  the  machine  would 
operate  as  a  three-phase  motor. 
If  points  AHF  and  /  were  used 
it  would  be  two-phase ;  if  AEBFC 
and  G,  six-phase;  while  if  con- 
nection were  made  to  all  twelve 
points  the  machine  would  be 
twelve-phase.  If  operated  on  any 
of  these  numbers  of  phases,  a 
rotating  magnetic  field  would  be 
set  up.  Each  of  the  coils  would 
therefore  generate  a  counter 
e.m.f.,  the  phase  of  which  would 
depend  upon  its  position  on  the 
stator,  and  there  would  exist  a 

three-phase  e.m.f.  at  the  three-phase  terminals,  a  two-phase  one  at 
the  corresponding  points,  etc.  No  matter  what  the  number  of  phases 
of  the  current  applied  it  would  be  possible  to  take  off  two,  three,  six,  or 
twelve-phase  current  from  the  appropriate  terminals. 

In  general,  it  would  not  be  advisable  to  use  an  induction  machine  in 
this  manner  to  transform  the  number  of  phases,  since  for  this  purpose  a 
number  of  transformers,  connected  in  well-known  ways,  would  be 
cheaper  and  better.  In  two  cases,  however,  this  arrangement  may  be 
advantageous.  It  can  be  readily  shown  that  it  is  not  possible  by  any 


FIG.  36.— E.M.F.  Relations  of  Various 
Coils  of  an  Induction  Motor. 


68  THE  INDUCTION  MOTOR 

combination  of  transformers  to  change  from  a  polyphase  system  to  single- 
phase,  and  have  the  currents  of  the  polyphase  system  balanced.  The 
energy  will  in  all  cases  be  supplied  as  single-phase  energy  from  the  gen- 
erator. In  general  the  power  in  single-phase  circuits  falls  to  zero  four 
times  in  each  cycle.  The  power  supplied  by  the  polyphase  line  must 
likewise  fall  to  zero,  unless  some  means  of  storing  energy  in  the  system 
is  provided.  The  kinetic  energy  of  the  rotor  of  an  induct'on  motor, 
used  as  a  phase  converter,  supplies  a  means  of  storing  this  energy.  The 
angular  velocity  of  the  rotor  is  not  constant,  but  varies  during  the 
revolution.  The  machine  takes  energy  in  approximately  equal 
amounts  from  all  the  phases  of  the  supply  system.  When  no  energy 
is  demanded  by  the  single -phase  system,  the  motor  is  being  accel- 
erated, and  energy  is  being  stored  as  kinetic  energy  in  the  rotor. 
During  the  parts  of  the  cycle  when  the  energy  demand  of  the  single- 
phase  system  is  heaviest,  the  rotor  is  retarded  and  gives  up  a  part  of 
its  kinetic  energy. 

The  induction  machine  may  also  be  used  in  the  opposite  manner  to 
transform  from  single-phase  to  any  polyphase  system.  When  operating 
near  synchronism  on  a  single-phase  sytsem,  provided  the  motor  has  a 
low-resistance  rotor,  a  rotating  magnetic  field  will  be  set  up,  .and  the 
magnitude  of  this  field  will  be  nearly  constant  in  all  positions.  Hence  a 
polyphase  e.m.f.  will  exist  at  the  terminals  of  the  corresponding  windings. 
The  voltages  of  the  polyphase  windings  will  not  be  exactly  balanced 
under  load,  but  will  be  nearly  so. 

An  example  of  an  application  where  this  principle  might  be  used  to 
advantage  is  in  the  case  of  a  plant  where  it  is  desired  to  introduce  motors 
to  drive  the  machinery,  and  only  single -phase  energy  is  available.  Of 
course  this  condition  might  be  met  by  the  use  of  single-phase  motors. 
As  is  well  known,  however,  such  motors  are  very  costly,  compared  with 
polyphase  motors  of  the  same  rating.  Moreover,  in  many  cases  there  is 
a  strong  possibility  that  the  supply  will  later  be  changed  to  polyphase. 
Under  these  circumstances,  it  might  be  well  to  install  standard  three- 
phase  motors,  starters  and  wiring.  The  three-phase  supply  lines  could 
then  at  any  future  time  be  connected  directly  to  the  wiring,  and  the  plant 
operated  three-phase. 

During  the  time  when  single-phase  energy  alone  was  available,  it 
would  be  necessary  to  add  a  phase-splitting  device  as  indicated  in  Fig.  37. 
This  can  be  constructed  as  shown  by  connecting  a  resistor  and  an 
reactor  in  series.  It  need  only  be  large  enough  to  start  one  of  the  three- 
phase  motors  unloaded.  It  may  then  be  disconnected  from  the  line,  and 


THE  INDUCTION  GENERATOR 


69 


the  one  motor  in  operation  will  generate  such  an  e.m.f.  as  to  cause  the 
third  wire  to  assume  the  proper  phase  relation  to  the  other  two,  and  form 
with  them  nearly  a  true  three-phase  system.  The  other  motors  may 
then  be  started  as  three-phase  motors. 


1  [r-Single  Phase  Supply 

5 

j 

i 

D  - 

1 

Phase 

Splitter 

111                   HI 

in            in 

Starter                     Starter 

Motor 


FIG.  37. — Connections  for  Operating  Three-phase  Induction  Motors  on  Single-phase 
Circuits. 


A  peculiarity  of  this  system  is  that  while  the  pull-out  point  of  all  the 
motors  if  loaded  each  in  proportion  to  its  rating,  would  correspond  to  the 
single-phase  rating  of  the  motors,  the  pull  out  point  of  any  single  motor, 
provided  the  other  motors  are  lightly  loaded,  is  practically  the  same  as 
its  pull-out  point  on  a  three-phase  circuit.  This  property  may  be  of 


Single  Phase 
Supply 


Three 
Phase 
Load 


FIG.  38. — Connections  of  Phase  Converter. 

great  value  in  cases  where  the  motors  are  subject  to  heavy  momentary 
overloads. 

To  test  this  action  of  an  induction  machine,  connections  were  made 
as  in  Fig.  38.  A  balanced  three-phase  load  was  applied  to  the  three- 
phase  line  and  the  voltage  between  the  lines  measured  for  various  values 
of  the  current.  The  results  are  plotted  in  Fig.  39.  The  motor  used 


70 


THE  INDUCTION  MOTOR 


had  a  three-phase  rating  of  7^  h.p.,  60  cycles  three-phase,  1200  rev.  per 
min.  220  volts.  Its  full-load  current,  operated  three-phase,  was  approxi- 
mately 21  amperes. 


210 
200 
If  HI 
1M) 

J.L 

lie 

A1  to 
A  *" 

Q 

_ 

*•* 

-\. 

>- 

\ 

" 

~\ 

"-^. 

^ 

KV. 

130 
I'.'O 
11(1 
10" 

M 
80 

TO 
fill 
5-1 
IB 

n 
00 

10 

""" 

^ 

^ 

•'• 

-», 

3 

"**• 

. 

Balanced  Amperes  per  Phase 

0     1     2     3     4     5     6     7     8     9    10    11   12    13    14    15    16   17   18    19   20 
FIG.  39. — Curves  of  Regulation  of  Phase  Converter. 


USE  OF  AN  INDUCTION  MACHINE  AS  A  VOLTAGE  BALANCER 
As  a  corollary  of  the  use  just  described,  it  will  be  readily  seen  that  if 
an  induction  machine  be  operated  from  a  polyphase  line,  the  voltages  of 
which  are  unbalanced,  that  it  will  act  to  restore  to  a  certain  extent  the 
correct  voltage  relation.  In  fact  the  case  just  described  is  merely  an 
extreme  case  of  unbalanced  voltage.  We  may  consider  the  system  as  a 
three-phase  system  in  which  the  voltage  between  either  of  the  two  outside 
lines  and  the  third  line  is  indefinite;  that  is,  it  may  be  at  the  potential  of 
either  of  the  other  two  lines.  By  starting  the  induction  machine,  it  is 
caused  to  assume  a  certain  definite  phase  relation  to  the  other  two 
lines. 


THE  INDUCTION  GENERATOR  71 

In  a  similar  way,  if  an  induction  machine  is  operated  on  an  nnbal- 
anced  circuit,  it  will  tend  to  take  most  of  its  energy  from  the  phases  the 
voltages  of  which  are  high,  and  little  or  none  from  those  which  are  low- 
In  fact,  it  may  readily  happen  that  it  even  returns  energy  to  the  low- 
voltage  circuits,  and  of  course  takes  an  extra  amount  of  energy 
from  the  high-voltage  circuits.  Thus  the  machine  may  be  acting 
at  the  same  time  as  a  motor  and  as  a  transformer,  taking  energy 
from  certain  heavily  loaded  circuits  and  transferring  it  to  others 
more  lightly  loaded. 


CHAPTER  VI 
VARIABLE  SPEED  INDUCTION  MOTORS 

IT  is  in  regard  to  its  adaptability  to  variable  speed  work,  that  the 
induction  motor  suffers  most  severely  in  comparison  with  the  direct-cur- 
rent motor.  In  fact  it  may  be  said  at  once  that  the  induction  motor  is 
distinctly  inferior  in  this  respect.  For  example  a  shunt-wound,  direct- 
current  motor,  by  the  simple  addition  of  a  shunt  field  rheostat,  may  be 
made  to  operate  through  a  considerable  range  of  speed.  The  efficiency 
at  any  of  these  speeds  will  be  approximately  the  same,  and  the  speed  for 
any  adjustment  of  the  field  rheostat  will  remain  practically  constant 
irrespective  of  the  load  applied.  On  the  other  hand,  for  certain  pur- 
poses, such  as  railway  operation,  a  motor  which  automatically  slows 
down  under  load  is  desired.  For  this  purpose  the  direct-current  series 
motor  is  excellently  adapted.  There  is  not  available  on  the  market  at 
the  present  time  any  polyphase  induction  motor  which  has  either  of 
these  two  characteristics.  It  is  true  that  by  the  addition  of  a  commuta- 
tor, and  the  provision  of  a  polyphase  regulating  transformer,  it  is  possi- 
ble to  build  a  motor  which  to  a  certain  extent  corresponds  to  the  adjusta- 
ble speed  shunt-wound  direct-current  motor,  but  the  expense  of  the 
necessary  additions,  together  with  the  complication  involved,  have  so 
far  prevented  its  commercial  application. 

The  fundamental  difference  in  characteristics  is  due  to  the  fact  that 
the  speed  of  a  direct-current  motor  depends  upon  the  voltage  and  voltage 
relations,  while  the  speed  of  an  induction  motor  depends  primarily  upon 
frequency.  The  voltage  of  a  circuit  is  easily  changed,  but  it  requires 
expensive  and  inefficient  apparatus  to  change  the  frequency. 

The  simplest  way,  and  the  one  most  commonly  used  to  obtain  speed 
variation  in  an  induction  motor,  is  to  provide  means  of  varying  the 
resistance  in  the  rotor  circuit.  This  is  usually  done  by  providing  a 
wound  rotor  with  slip  rings,  and  an  adjustable  external  resistance.  The 
insertion  of  resistance  increases  the  slip  and  consequently  lowers  the 
speed. 

It  will  be  seen  at  once  that  this  is  an  inefficient  method  of  speed  con- 
trol. The  rotor  as  was  explained  on  page  32,  always  requires  an 

72 


VARIABLE  SPEED  INDUCTION  MOTORS 


73 


amount  of  power  equal  to  the  synchronous  watts,  i.e.,  to  the  power  the 
rotor  would  develop  if  it  were  operating  at  synchronous  speed  with  the 
torque  that  it  is  developing.  Thus  if  enough  resistance  is  inserted  to 
reduce  the  speed  to  half  the  synchronous  speed,  the  rotor  must  receive  an 
amount  of  power  double  that  which  it  is  developing.  The  efficiency  of 
the  rotor  is  therefore  50  per  cent,  and  the  efficiency  of  the  whole  machine 
is  even  less  than  this.  If  an  attempt  is  made  to  operate  at  say  10  per 
cent  of  full  speed,  the  efficiency  will  likewise  be  less  than  10  per  cent. 

There  is  also  another  difficulty.     The  speed  for  any  setting  of  the 
rotor  resistance  will  vary  with  the  load.     This  is  especially  true  if  the 


FlG.  40. — Speed  Torque  Curves  of  Wound-rotor  Induction  Motor. 


speed  reduction  at  full  load  is  considerable.  In  Fig.  40  are  shown 
speed  -torque  curves  for  various  secondary  resistances  of  a  typical  induc- 
tion motor.  The  equation  for  the  torque  was  derived  on  page  25  and 

sE2R 
was  found  to  be  D  =—  -         —  —  .  In  this  case,  E,  the  applied  voltage,  L, 


the  inductance  of  the  motor,  and  to  (equal  to  211  times  the  line  frequency) 
are  constant.  Assuming  then,  any  value  of  R,  the  sum  of  the  internal 
and  external  resistance,  we  can  select  various  values  of  s,  the  slip,  and 
solve  for  the  torque  D.  Instead,  however,  of  plotting  the  slip,  we  have 
used  instead  the  speed  =  1—5.  The  curve  A  represents  the  case  where 
the  rotor  resistance  is  a  minimum,  i.e.,  the  slip  rings  are  short-circuited. 
In  curve  BCDE,  etc.,  we  have  constantly  increasing  resistances.  The 
maximum  torque  is  the  same  in  all  cases,  but  the  speed  at  which  it  is 


74  THE  INDUCTION  MOTOR 

developed  changes  with  the  resistance.     This  will  be  readily  seen  if 


we  write  the  above  equation  in  the  form,  D=         .         —  .      It  will 


be  evident  that  for  any  slip  s,  it  will  be  possible  to  select  a  value  of  R 
such  that  any  desired  value  of  the  fraction  will  be  obtained.  Hence 
any  possible  value  of  the  torque  D  may  be  secured  at  any  slip  by  a 
suitable  choice  of  the  resistance.  The  dotted  parts  of  curves  E  and  F 
indicates  that  to  develop  the  maximum  torque,  the  speed  must  be 
negative,  that  is,  the  motor  must  be  rotated  in  the  reverse  direction. 
The  principles  upon  which  these  facts  depend  have  been  fully  explained, 
and  need  not  be  repeated  here. 

CONTROL  DEVICES 

The  wound-rotor  machine  with  slip  rings  may  be  employed  for  either 
of  two  purposes  ;  to  enable  the  speed  of  the  machine  to  be  var'ed,  or  to 
give  better  starting  conditions.  The  methods  employed  in  the  two  cases 
are  essentially  the  same,  with  the  obvious  difference  that  in  the  case  of 
the  device  intended  for  starting  duty  only,  much  less  resistance  material 
is  necessary  than  in  the  case  where  the  resistance  is  in  circuit  constantly. 
For  reasons  already  given,  the  rotors  of  wound-rotor  machines  are 
always  three-phase.  The  secondary  resistors  may  be  connected  either  in 
star  or  in  delta. 

A  typical  motor  with  its  controller  and  resistors  are  shown  in  Fig.  41. 
This  controller  in  addition  to  varying  the  resistance  in  the  rotor  circuit 
is  fitted  with  contacts  to  open  two  of  the  three  primary  circuits.  Thus 
the  current  supply  to  the  stator  is  interrupted  when  the  controller  handle 
is  in  the  neutral  position.  The  contacts  are,  moreover,  so  arranged  that 
the  connections  of  the  two  primary  leads  to  the  stator  are  reversed  when 
the  handle  is  moved  from  the  right  to  the  left  of  the  central  position. 
This  has  the  effect  of  reversing  the  direction  of  rotation  of  the  motor. 
Moving  the  handle  further  from  the  central  position  cuts  out  more  of  the 
resistance  in  the  rotor  circuit. 

In  arranging  the  contacts  of  the  controller  circuits  it  is  impracticable 
to  have  the  secondary  resistance  balanced  at  all  times.  Suppose  for 
example  it  was  desired  to  have  eight  running  points.  If  the  phases  were 
to  be  kept  balanced  at  all  points,  it  would  be  necessary  to  have  seven 
contacts  connected  to  each  of  the  three  resistors  or  twenty-one  in  all. 
By  arranging  taps  as  in  Fig.  42,  the  necessary  number  is  reduced  to 


VARIABLE  SPEED  INDUCTION  MOTORS 


75 


seven.  The  various  steps  are  secured  by  connecting  the  points  2,  3, 
4,  etc.,  successively  to  the  neutral  point  i.  The  resistances  may  be 
so  arranged  that  the  circuits  are  balanced  on  any  one  of  the  points 


FIG.   41. — Rotor   and   Induction   Motor   with   Controller.     Built   by   Fairbanks, 
Morse  and  Co. 


desired.  Thus  the  point  of  balance  may  be  the  first  point,  or  it  may  be 
at  the  point  of  maximum  torque.  The  unbalancing  at  the  other  points 
has  so  little  effect  as  to  be  unobjectionable. 


FIG.  42. — Diagram  of  Starting  Resistances. 

A  rheostat  built  by  the  Cutler-Hammer  Manufacturing  Co.  and 
intended  for  starting  purposes  only  is  shown  in  Fig.  43.  The  con- 
struction will  be  obvious  from  the  illustration.  In  this  case  also  the 


70 


THE  INDUCTION  MOTOR 


starting  resistances,  are  unbalanced.  There  are  only  two  resistors 
connected  in  open  V  with  the  three  leads  connected  to  the  three  points 
of  the  V.  This  rather  serious  unbalancing  seems  to  make  but  little 
reduction  in  the  starting  torque.  Starters  of  the  carbon  block  type  are 


FIG.  43. — Starter  for  Wound  Rotor  Induction  Motor 
Built  by  Cutler-Hammer  Mfg.  Co. 


also  built  for  this  service  as  well  as  for  regulating  duty,  and.  these  have 
the  advantage  of  keeping  the  circuits  balanced  at  all  times. 


SPEED  VARIATION  BY  CASCADE  CONNECTION 

The  connection  shown  in  Fig.  44  is  known  as  the  cascade  connection, 
or  as  connection  in  concatination.  Consider  for  example  that  both 
motors  in  the  figure  have  the  same  number  of  turns  on  the  secondary  as 
on  the  primary,  and  that  both  are  wound  for  three  phases.  Let  motor 
A  be  operating  alone,  and  let  us  assume  that  sufficient  resistance  has 
been  connected  in  the  rotor  circuit  so  that  it  is  operating  at  half  of  the 
synchronous  speed.  If  the  rotor  were  at  rest,  the  secondary  voltage, 
neglecting  the  slight  drop  due  to  leakage  and  to  the  loss  in  the  resistance, 
would  be  the  same  as  that  of  the  primary.  When  operating  at  half 
synchronism,  the  secondary  voltage  will  be  just  half  that  of  the  primary, 
since  the  cutting  will  be  half  as  rapid.  Moreover,  the  frequency  of  the 
secondary  current  will  be  half  that  of  the  primary.  It  will  occur  at  once 
to  the  reader  that  we  might  use  this  current  to  operate  another  motor 


VARIABLE  SPEED  INDUCTION  MOTORS 


77 


instead  of  wasting  it  in  resistance.  This  can  readily  be  done  by  connect- 
ing the  two  motors  as  shown  in  Fig.  44. 

It  is  necessary,  however,  that  the  two  motors  be  connected  mechan- 
ically as  well  as  electrically.  This  connection  may  take  the  form  of  a 
belt  and  pulleys,  it  may  consist  of  gears,  or  the  two  motors  may  be 
mounted  directly  on  the  same  shaft.  The  last  is  obviously  the  best  when 
conditions  permit.  If  this  mechanical  connection  were  not  present, 
the  speeds  would  be  the  same  only  when  the  torques  were  approximately 
equal.  With  lighter  torque  on  the  first  motor  it  would  speed  up  and  the 
second  one  slow  down,  and  vice  versa.  The  sum  of  the  two  speeds  would 
under  all  circumstances  be  approximately  constant. 

If,  however,  the  two  are  mechanically  connected  as  indicated,  the  set 
will  under  all  loads  run  at  nearly  a  constant  soeed.  Assume,  for  example, 


FIG.  44. — Induction  Motors  in  Cascade 


that  the  two  motors  have  the  same  number  of  poles,  and  that  they  are 
direct-connected  or  connected  by  a  one  to  one  gearing.  The  set  will 
then  tend  to  run  at  half  synchronous  speed.  That  this  is  so  will  be 
readily  seen.  Suppose  for  example  that  the  speed  is,  at  a  given  instant, 
higher  than  this.  The  frequency  of  the  secondary  current  of  the  motor  A 
will  be  less  than  half  the  applied  frequency,  while  the  frequency  of 
rotation  of  the  second  motor  will  be  greater  than  half  of  the  applied 
frequency.  The  machine  B  will  therefore  act  as  an  induction  gener- 
ator, and  will  tend  to  establish  current  in"the  rotor  of  A  against  its 
e.m.f.  A  will  therefore  likewise  establish  current  in  the  supply  line,  and 
the  whole  set  will  act  as  an  induction  generator.  The  set  will  there- 
fore slow  down  until  the  secondary  frequency  of  A  is  the  same  as  the 
frequency  corresponding  to  the  rate  of  revolution  of  B.  The  synchro- 
nous speed  of  the  set  is  therefore  equal  to  half  the  synchronous  speed  of 
either  one  of  the  motors  alone. 

The  speed  of  the  set  when  the  number  of  poles  of  the  two  motors  is 


78  THE  INDUCTION  MOTOR 

not  the  same  is  readily  deduced.  As  previously  stated,  it  is  necessary 
that  the  speed  of  the  motor  B  must  be  such  that  the  frequency  corre- 
sponding to  its  speed  of  rotation  is  approximately  the  same  as  the  fre- 
quency of  the  secondary  of  A.  It  will  be  seen  that  this  leads  to  the  rule 
that  the  synchronous  speed  of  the  set  when  the  two  machines  are  direct- 
connected  will  be  that  corresponding  to  the  speed  of  a  single  machine 
having  as  many  poles  as  the  sum  of  the  poles  of  the  two  motors.  Thus 
if  the  motors  have  respectively  six  and  ten  poles,  the  individual  syn- 
chronous speeds  on  a  6o-cycle  circuit  will  be  1200  and  720  rev.  per  min. 
The  speed  of  the  combined  set  will  be  that  of  a  i6-pole  machine  or  450 
rev.  per  min.  With  such  a  set  we  therefore  have  available  three  speeds, 
and  since  these  are  obtained  without  the  use  of  resistors,  the  set  is  work- 
ing at  high  efficiency  at  all  of  them. 

Considering  again  the  connection  shown  in  Fig.  44,  let  the  two  motors 
be  exactly  alike,  and  let  the  rotors  have  the  same  number  of  turns  as  the 
stators.  They  may  then  be  operated  singly  from  the  primary  supply,  or 
they  may  both  be  supplied  at  the  primary  voltage  and  frequency,  thus 
giving  twice  the  capacity  of  one  motor,  or  they  may  be  operated  in  cas- 
cade at  half  the  former  speed.  As  was  pointed  out,  when  operating  in 
cascade,  the  machine  B  will  be  supplied  with  current  at  half  voltage  and 
half  frequency.  This  voltage  is,  moreover,  correct  for  this  frequency. 
That  this  is  so  will  be  apparent  if  we  consider  that  since  the  voltage  and 
the  frequency  are  both  half  of  normal,  the  flux  in  the  stator  will  be  of  the 
normal  value.  This  is  as  it  should  be,  since  if  the  flux  were  less,  we 
should  not  get  the  maximum  output  from  the  motor,  and  if  it  were  much 
greater  we  should  be  in  danger  of  heating  or  of  undue  losses. 

To  understand  the  action  of  the  set  when  operating  in  cascade,  let  us 
consider  the  action  of  the  motor  B  when  operating  at  normal  voltage 
and  frequency  and  again  when  operating  at  half  voltage  and  half  fre- 
quency. Fig.  45  shows  the  respective  circle  diagrams  under  these  two 
conditions.  At  will  be  seen,  the  current  circles  are  almost  the  same. 
That  the  magnetizing  current  will  be  the  same  is  apparent,  since  the  flux 
remains  the  same,  and  the  magnetizing  current  is  inedpendent  of  the 
voltage  or  of  the  frequency  (see  page  106).  The  locked  current  for 

zero  motor  resistance  will  also  be  unchanged  since  this  current  is  equal  to 

•p 
—  .  The  e.m.f .  acting  on  the  rotor  is  evidently  half  at  the  lower 

voltage  and  frequency,  but  the  frequency  being  half  also  and  the  other 
quantities  remaining  unchanged,  the  current  will  be  the  same.  The  only 
difference  in  the  circle  comes  from  the  fact  that  the  iron  losses  are  approx- 


VARIABLE  SPEED  INDUCTION  MOTORS 


7!) 


imately  half  as  great  at  the  lower  frequency  and  consequently  the  circle 
is  somewhat  lowered. 

While  it  is  true  that  the  circle  is  practically  unchanged,  the  applied 
e.m.f.  is  only  half  that  of  the  line,  and  consequently  the  outupt  of  the 
motor  is  only  half  as  great  as  its  output  when  operating  at  primary  voltage 
and  frequency.  This  might  have  been  anticipated  from  the  fact  that 
the  motor  is  operating  at  half  speed.  Moreover,  since  the  motor  A  is 
likewise  operating  at  half  speed,  and  since  on  account  of  the  equality  of 
the  turns  on  the  primary  and  secondary,  its  primary  current  is  practically 
equal  to  that  of  B,  its  output  will  also  be  equal  to  half  its  output  at  full 
speed.  The  combined  output  of  the  set  is  then  equal  to  the  output  of  one 
machine  at  full  speed,  or  is  equal  to  half  that  of  the  two  machines  at  full 
speed.  From  this  it  will  be  readily  seen  that  the  torque  of  each  motor 


Cycles 
Cycles 


FIG.  45. — Circle  Diagram  of  6o-Cycle  and  3o-Cycle  Motors. 


is  the  same  at  the  low  speed  as  at  the  high  speed.  The  action,  both  as 
regards  the  output  at  the  half  speed,  and  as  regards  the  torque  avail- 
able, is  similar  to  that  of  two  direct-current  shunt  motors  operated 
with  the  armatures  in  series  and  in  parallel,  the  fields  having  at  all 
times  full  voltage  applied. 

POWER-FACTOR 

It  is  unfortunately  true  that  the  power-factor  of  two  motors  operated 
in  cascade  is  lower  than  that  of  either  of  the  motors  alone.  That  this  is 
so  is  evident  when  it  is  remembered  that  both  motors  are  operating  with 
full  flux  in  the  stators.  They  wrill  therefore  require  the  same  volt- 
amperes  excitation  as  though  each  were  operating  alone.  The  total 
input  to  the  set  is,  however,  only  half  of  that  which  it  would  take  if  both 
machines  were  directly  connected  to  the  line.  The  percentage  of  watt- 


80  THE  INDUCTION  MOTOR 

less  current  is  therefore  twice  as  great  when  the  machines  are  operating  in 
cascade,  and  the  power-factor  is  correspondingly  reduced. 

It  should  also  be  noted  that  in  the  above,  it  was  assumed  that  the 
motor  B  was  supplied  with  current  at  half  the  primary  voltage  and  fre- 
quency. As  a  matter  of  fact,  both  the  voltage  and  the  frequency  fall 
below  these  values.  The  output  of  the  second  motor  is  therefore  some- 
what less  than  the  above  considerations  would  tend  to  show. 

The  torque  at  starting  is  of  importance.  As  has  been  shown,  the 
starting  torque  in  synchronous  watts  is  equal  to  the  loss  in  the  rotor 
circuit.  When  the  two  machines  are  connected  in  cascade  and  the  set 
is  at  rest,  the  first  machine  acts  as  a  transformer  to  supply  current  to  the 
second  machine.  The  second  machine  will  receive  power  at  the  full 
voltage  and  frequency,  assuming  that  the  first  machine  acts  as  a  perfect 
transformer.  It  will  therefore  develop  its  full  torque  in  the  same  manner 
as  though  it  were  connected  to  the  line.  Since,  however,  there  is  some 
drop  in  voltage  the  torque  will  be  somewhat  reduced. 

At  the  same  time  the  first  machine  is  developing  torque.  This  torque 
in  synchronous  watts  will  be  equal  as  before  to  the  power  developed  in 
its  rotor  circuit.  This  power  is  the  same  as  the  power  in  the  rotor  cir- 
cuit of  the  second  machine,  plus  the  losses  in  the  rotor  of  A  and  in  the 
stator  of  B.  The  torque  of  the  machine  A  is  therefore  somewhat  greater 
tharf  that  of  B.  The  torque  in  foot-pounds  of  the  machines  in  cascade 
is  therefore  approximately  twice  that  of  one  of  the  set,  or  it  is  equal  to 
that  which  would  be  developed  by  both  of  them,  operated  directly  on  the 
primary  circuit.  The  power  taken  from  the  circuit  if  they  were  operated 
in  this  way  would,  however,  be  twice  as  great  as  with  the  cascade  con- 
nection. 

VARIABLE  NUMBER  OF  POLES 

A  motor  may  be  constructed  with  two  distinct  stator  windings,  each 
wound  for  a  different  number  of  poles.  If  the  rotor  is  of  the  squirrel- 
cage  variety,  the  motor  will  operate  at  either  of  the  two  speeds  corre- 
sponding to  the  two  sets  of  poles.  If  a  wound  rotor  is  used,  it  is  necessary 
that  two  windings  be  employed  on  it  also,  since  a  ten-pole  stator,  for 
example,  would  produce  no  current  in  a  six-pole  rotor.  The  squirrel-cage 
rotor  has  no  definite  number  of  poles,  but  automatically  adjusts  itself  to 
the  number  of  stator  poles. 

Such  a  motor  will  have  different  outputs  at  the  two  speeds.  These 
outputs  will  be  proportional  respectively  to  the  two  speeds.  In  this  it  is 
similar  to  the  two  motors  operated  in  cascade.  The  principal  difficulty 


VARIABLE  SPEED  INDUCTION  MOTORS  81 

in  constructing  such  motors  is  the  necessity  of  using  very  large  slots  to 
contain  the  two  windings.  This  in  turn  necessitates  a  greater  outside 
diameter  of  the  motor,  and  on  account  of  one  winding  being  far  from  the 
rotor  surface,  causes  the  motor  to  have  a  large  leakage  factor,  and  a  low 
power-factor.  It  also  reduces  the  pull-out  point,  and  to  a  certain  extent, 
the  efficiency. 

An  interesting  variation  of  this  plan  is  to  wind  the  motor  with  one  set 
of  coils  which  are  of  rather  short  pitch  for  the  smaller  number  of  poles. 
These  coils  are  not  permanently  connected  together,  but  are  so  arranged 
that  by  means  of  a  special  switch  the  coils  may  be  so  connected  as  to 
change  the  number  of  poles. 

The  objection  to  this  scheme  is  the  fact  that  the  pitch  is  not  in  general 
the  best  for  either  number  of  poles,  and  what  is  of  greater  impor- 
tance, the  switch  is  necessarily  very  complicated.  It  will  be  readily  seen 
that  with  such  a  switch  a  short  or  an  open  circuit  might  readily  occur. 
The  former  would  probably  lead  to  burning  out  a  coil;  the  latter  would 
cause  the  motor  to  operate  single  phase.  It  is,  moreover,  apparent  that 
the  two  windings  will  not  be  adapted  to  the  same  voltage.  If  the  flux 
density  is  kept  the  same  in  each,  and  if  the  number  of  coils  in  series  is  the 
same  in  each,  the  voltage  will  be  proportional  to  the  respective  speeds. 
Of  course,  in  certain  cases,  this  may  be  taken  care  of  in  the  switching 
device.  For  example,  if  the  speeds  are  two  to  one,  twice  as  many  coils 
may  be  connected  in  series  for  the  lower  speed.  The  disadvantages  of 
both  of  these  schemes  are  such  that  they  are  rarely  used. 

Besides  the  method  of  obtaining  two  different  numbers  of  poles,  by 
utilizing  two  separate  windings,  various  connections  have  been  proposed 
by  means  of  which  it  is  possible  to  change  the  points  of  connection  of  the 
circuit  to  a  single  winding  so  as  to  produce  the  same  effect.  These  are 
applicable  only  in  cases  where  the  number  of  poles  with  one  connection 
is  twice  that  with  the  other. 

In  the  practical  application  of  this  method,  the  coils  are  of  such  a 
span  as  to  give  full  pitch  with  the  larger  number  of  poles,  and  conse- 
quently the  pitch  is  half  of  full  pitch  when  the  smaller  number  of  poles 
is  used.  The  connections  are  so  made  that  when  the  motor  is  operating 
with  the  greater  number  of  poles,  half  of  the  poles  are  consequent  poles, 
and  the  windings  as  before  noted  are  of  full  pitch.  To  obtain  the 
smaller  number  of  poles,  it  is  necessary  to  conduct  the  current  to  what 
was  the  center  of  a  winding.  To  preserve  approximately  the  same  flux 
density  at  the  higher  speed,  it  is  necessary  at  the  same  time  to  change  the 
connections  of  the  coils  from  mesh  to  star,  or  for  the  larger  number  of 


82  THE  INDUCTION  MOTOR 

poles  the  connection  is  series  mesh,  and  for  the  smaller,  parallel  star. 
The  external  connections  by  which  this  is  done  are  shown  in  Fig.  46.  A 
full  connection  diagram  for  this  method  of  changing  poles  will  be  found 
in  "  Electric  Motors  "  by  H.  M.  Hobart,  page  571  2nd  edition. 

If  the  machine  is  one  with  a  wound  rotor,  it  is  of  course  necessary  in 




Eight  Poles  x  Four  Poles 

FIG.  46. — Connection  for  Changing  the  Number  of  Poles. 

applying  any  of  the  methods  in  which  the  number  of  pole  is  changed, 
to  alter  the  number  of  rotor  poles  at  the  same  time  that  those  in  the  stator 
are  changed.  This  involves  the  use  of  five  collector  rings  if  two  separate 
windings  are  used,  or  of  six  rings  if  the  winding  just  described  is 
employed.  If  a  squirrel-cage  winding  is  used,  it  of  course  adapts  itself 
to  any  number  of  poles. 

THE  COMMUTATOR  TYPE  POLYPHASE  INDUCTION  MOTOR 

The  induction  motor,  as  usually  constructed,  especially  if  of  the 
squirrel-cage  type,  is  almost  ideally  simple.  In  this  respect  it  is  prob- 
ably superior  to  any  other  device  for  the  conversion  of  any  form  of  energy 
into  mechanical  work.  As  has  been  pointed  out,  however,  it  is  inferior 
to  the  direct-current  motor  in  several  respects,  the  most  important  of 
which  are  the  fact  that  the  speed  is  not  readily  changed  in  an  efficient 
manner,  and  the  fact  that  the  power-factor  is  always  materially  less  than 
unity.  Both  of  these  objections  can  be  overcome,  if  we  are  willing  to 
sacrifice  some  of  the  simplicity  of  the  usual  motor. 

Fig.  47  is  a  representation  of  a  direct-current  armature.  The  wind- 
ing shown  is  a  simplex  lap.  Any  other  of  the  direct-current  windings 
might  have  been  used.  Imagine  this  armature  placed  in  an  ordinary 
induction  m®tor  stator,  instead  of  the  usual  rotor.  If  the  commutator 
were  short  circuited,  by  winding  a  wire  around  it  or  otherwise,  the 


VARIABLE  SPEED  INDUCTION  MOTORS 


83 


machine  would  operate  in  essentially  the  same  manner  as  a  squirrel-cage 
motor.  The  currents  in  the  rotor  would  be  as  shown  in  Fig.  7. 

Suppose  that  instead  of  short-circuiting  the  rotor  commutator  in  this 
manner,  we  make  permanent  connection  to  two  of  the  commutator  bars 
as  A  and  B  and  pass  in  a  direct-current.  This  would  of  course  have  to 
be  done  by  means  of  slip  rings  and  brushes.  Under  these  conditions  the 
machine  would  become  a  synchronous  motor. 

The  path  of  the  currents  in  the  rotor  would  be  as  shown  by  the  arrows. 
It  will  be  seen  that  this  gives  rise  to  distinct  bands  of  conductors,  each 
band  carrying  current  in  only  one  direction.  These  currents  act  to 
cause  poles  in  the  rotor  in  the  positions  shown,  the  observer  being  sup- 
posed to  be  looking  out  from  the  inside  of  the  rotor.  As  was  previously 
explained,  it  is  perhaps  better  from  the  theoretical  standpoint  to  con- 


IA|     I     I     I     I     |B| 


|     |     |     |     |     | 


FIG.  47.  —  Currents  in  Armature  of  Direct  Current  Motor. 

sider  the  reaction  of  the  rotor  currents  and  the  stator  magnetism,  but  the 
concept  of  poles  is  very  useful  in  gaining  a  clear  physical  idea  of  the 
actions  involved.  It  will  be  seen  at  once  that  the  "  poles  "  of  the  stator 
will  attract  those  of  the  rotor  and  carry  them  around  in  synchronism 
with  the  flux.  The  machine  has  thus  become  a  synchronous  motor. 
In  a  similar  manner,  a  motor  with  wound  rotor  may  be  converted  into  a 
synchronous  motor  by  passing  direct  current  in  through  one  of  the  slip 
rings  and  out  through  one  of  the  others.  The  third  ring  would  not  be 
used  at  all,  and  the  corresponding  section  of  the  winding  would  carry  no 
current. 

Fig.  48  represents  the  circle  diagram  for  a  synchronous  motor.  The 
diagram  is  not  strictly  correct,  as  the  effect  of  the  stator  resistance  has 
been  neglected,  but  it  is  sufficiently  close  for  our  purpose.  The  circle 
marked  /  may  be  taken  as  the  circle  diagram  of  an  induction  motor. 


84 


THE  INDUCTION  MOTOR 


It  will  be  recalled  that  the  reason  for  the  wattless  component  of  the  stator 
current  was  that  this  current  was  required  to  force  the  flux  across  the 
air-gap.  We  are  now,  however,  passing  direct  current  into  the  rotor,  and 
if  just  enough  is  supplied  to  force  the  required  flux  through  the  air-gap, 
this  magnetizing  component  of  the  stator  current  will  disappear.  The 
current  curve  for  the  circle  diagram  will  then  be  represented  by  the  circle 
II.  The  current  is  now  nearly  in  phase  with  the  e.m.f.  for  light  loads, 
or  the  power-factor  is  nearly  unity.  A  further  increase  in  the  direct  cur- 
rent in  the  rotor  gives  us  the  curve  72  or  73.  With  such  an  excitation,  the 
current  is  leading  for  most  loads,  and  lagging  for  very  heavy  loads  only. 
For  any  value  of  the  power  required,  we  can  pick  out  such  a  value  of  the 
field  current  as  to  make  the  power-factor  unity. 


FIG.  48. — Circle  Diagrams  of  Synchronous  Motor. 


Considering  again  the  direct-current  armature,  instead  of  using  slip 
rings  as  assumed,  let  us  place  on  the  commutator  three  brushes  for  each 
pair  of  poles.  If  the  armature  were  wave  wound,  of  course  only  three 
brushes  in  all  would  have  to  be  used.  These  brushes  should  be  spaced 
120  electrical  degrees  apart.  If  three-phase  current  be  passed  into 
these  brushes,  a  rotating  magnetic  field  will  be  set  up  in  precisely  the 
same  manner  as  would  be  the  case  if  the  current  were  passed  into  the 
stator.  Of  course  two-phase  current  could  be  used  with  four  brushes 
per  pair  of  poles. 

This  magnetic  field  will  rotate  in  space  with  synchronous  velocity, 
and  this  velocity  will  remain  the  same  irrespective  of  the  speed  of  the  rotor 
itself.  Imagine  now  that  three-phase  current  is  applied  to  the  stator 
and  at  the  same  time  to  the  rotor  through  the  brushes.  The  connections 
are  supposed  to  be  so  made  that  the  fields  rotate  in  the  same  direction. 
The  relative  position  of  the  poles  of  the  two  fields  will  depend  upon  the 


VARIABLE  SPEED  INDUCTION  MOTORS  85 

relative  position  of  the  taps  into  the  a  stator  winding  and  the  position  of 
the  brushes.  If  the  latter  are  so  set  that  the  two  fields  are  not  in  the 
same  line,  there  will  be  a  torque  between  the  fields,  and  provided  the  load 
is  not  too  great,  the  rotor  will  start  from  rest  and  attain  a  certain  speed. 
The  direction  of  rotation  may  be  the  same  as  the  direction  of  rotation 
of  the  two  fields,  or  the  reverse.  This  latter  direction  would,  however, 
not  be  used  in  practice. 

The  current  in  the  rotor  may  be  supplied  from  the  line  as  indicated 
above,  or  it  may  be  induced  in  the  rotor,  the  brushes  being  either  short 
circuited  or  connected  together  by  resistances.  In  this  case  the  motor 
will  operate  below  synchronism,  and  the  action  will  be  essentially  the 
same  as  that  in  the  ordinary  induction  motor,  the  speed  being  near  syn- 
chronism if  the  resistance  used  is  small,  and  at  a  lesser  speed  as  the  resist- 
ance is  increased.  If,  on  the  other  hand,  the  current  be  introduced  from 
the  line,  the  speed  will  be  greater  or  less  than  synchronism,  as  the  voltage 
is  applied  so  as  to  help  or  oppose  the  current  which  is  produced  by  gen- 
erated e.m.f.  Thus  assume  the  motor  to  be  operating  with  no  load  and 
the  brushes  short-circuited.  The  speed  will  be  very  near  synchronism. 
Imagine  a  small  e.m.f.  applied  from  the  line  so  as  to  increase  the  current 
through  the  armature.  The  increased  current  will  cause  an  increased 
torque  between  the  rotor  and  the  stator  and  the  speed  of  the  rotor  will 
increase.  This  increase  of  speed  will  cause  the  e.m.f.  generated  in  the 
rotor  to  decrease,  and  if  the  speed  is  increased  above  synchronism,  to 
reverse.  The  generated  e.m.f.  will  then  be  in  opposition  to  the  applied 
e.m.f.  and  will  tend  to  cut  down  the  current  in  the  rotor.  This  will  con- 
tinue until  the  difference  of  the  two  e.m.fs.  is  just  sufficient  to  maintain 
enough  current  to  supply  the  torque  required  to  maintain  the  rotation. 

If,  on  the  other  hand,  the  applied  e.m.f.  is  in  such  a  direction  as  to 
oppose  the  secondary  current,  this  current  will  be  lessened  or  reversed 
and  the  rotor  will  slow  down  until  the  current  again  reaches  the  proper 
value  to  maintain  the  rotation.  Thus  theoretically,  at  least,  we  could 
attain  any  speed  from  zero  up  to  the  limit  imposed  by  centrifugal  force. 
As  a  matter  of  fact,  as  will  be  explained  later,  it  is  not  practicable  to 
utilize  all  of  this  speed  range. 

It  will  be  noted  that  to  attain  the  above  speed  variation,  use  is  made 
of  various  voltages  applied  to  the  armature.  These  can  be  obtained  by 
the  use  of  an  auto-transformer  of  the  required  number  of  phases  or  by 
the  use  of  taps  brought  out  from  the  stator  winding  itself.  Some 
difficulty  would  be  experienced  in  this  latter  case,  since  the  winding 
must  act  as  an  auto-transformer  winding  at  the  same  time  that  it  acts  as 


86  THE  INDUCTION  MOTOR 

the  primary  of  the  motor.  Since  no  resistance  is  used  the  method  would 
be  efficient,  and  the  only  added  losses  would  be  those  due  to  the  auto- 
transformer  used. 

We  have  still  to  explain  how  the  power-factor  of  the  motor  may  be 
raised  to  unity,  or  the  motor  even  caused  to  take  a  leading  current.  To 
do  this,  we  may  add  three  more  brushes,  90  electrical  degrees  on 
the  commutator  from  the  main  or  power  brushes.  To  understand 
the  action,  assume  the  rotor  to  be  caused  to  rotate  at  synchronous  speed 
by  outside  power.  Imagine  the  brushes  to  be  on  the  commutator  as 
before,  but  shifted  half  a  pole  pitch,  so  that  the  rotating  field  produced  in 
the  rotor  coincides  with  that  in  the  stator  instead  of  differing  from  it  by 
90  degrees.  It  is  evident  that  since  the  two  fields  are  in  the  same  line, 
there  will  be  no  torque  between  them.  If  the  fields  act  to  magnetize  the 
structure  in  the  same  direction,  the  magnetizing  current  will  be  divided 
between  the  two  in  a  ratio  depending  upon  the  relative  values  of  the 
e.m.fs.  applied  to  them.  Thus  if  no  e.m.f.  be  impressed  on  the  rotor,  the 
stator  will  carry  the  magnetizing  current  as  in  the  ordinary  induction 
motor.  If,  on  the  other  hand,  no  e.m.f.  be  applied  to  the  stator  winding, 
the  rotor  will  carry  all  the  magnetizing  current.  Even  though  an  e.m.f. 
be  applied  to  the  stator,  the  rotor  may  still  carry  all  the  magnetizing 
current,  provided  the  voltage  applied  to  it  is  proportionately  greater  than 
that  applied  to  the  stator. 

The  question  will  occur  to  the  reader,  What  difference  does  it  make 
whether  the  magnetizing  current  is  carried  by  the  stator  or  the  rotor? 
It  would  make  no  difference  if  the  rotor  were  at  rest,  but  when  it  is  oper- 
ating at  synchronism,  it  can  be  readily  shown  that  the  rotor  inductance  is 
lower  than  when  it  is  at  rest,  and  hence  it  takes  a  current  more  nearly  in 
phase  with  the  applied  e.m.f.  and  consequently  at  nearly  unity  power- 
factor.  That  the  inductance  of  the  rotor  at  synchronism  will  be  small  is 
readily  apparent  if  we  consider  that  since  the  rotation  of  the  coils  is  syn- 
chronous with  the  changes  in  value  of  the  current,  the  current  in  any 
given  coil  will  never  entirely  reverse.  It  is  true  that  the  current  varies 
somewhat  in  value  in  each  coil,  and  consequently  there  will  be  some 
inductance,  but  it  will  be  less  than  would  be  the  case  if  the  coils  were  at 
rest.  We  may  look  at  the  matter  in  a  slightly  different  way  if  we  con- 
sider that  the  commutator  acts  to  change  the  current  supplied  into  a 
direct  current.  The  action  of  the  machine  is  then  essentailly  that  of  a 
synchronous  motor. 

It  will  be  evident  that  the  inductance  of  the  rotor  will  be  practically 
zero,  only  when  the  motor  is  operating  at  synchronism.  At  any  other 


VARIABLE  SPEED  INDUCTION  MOTORS 


87 


speed  it  will  be  necessary  to  apply  more  e.m.f .  to  the  brushes  to  maintain 
the  required  current  through  the  rotor.  The  compensating  action  will 
therefore  be  correct  for  only  one  speed,  unless  adjusted  for  each  change 


Line 


FIG.  49. — Connections  of  Variable  Speed  Induction  Motor. 

of  speed.  This  could  be  accomplished  by  interconnecting  the  auto- 
transformers  supplying  current  to  the  power  brushes  and  to  the  com- 
pensating brushes  in  such  a  manner  that  as  one  voltage  was  varied,  the 
other  would  be  varied  also  in  the  proper  manner.  As  can  be  readily 
seen,  this  leads  to  considerable  complication.  The  complete  connec- 
tions for  a  variable  speed  compensated  motor  are  shown  in  Fig.  49. 

If  it  is  not  essential  that  the 
motor  be  adapted  for  variable 
speed  work,  the  construction  can 
be  simplified  by  omitting  the 
power  brushes  and  connecting 
each  commutator  bar  to  its  neigh- 
bor by  a  small  resistor.  If  the 
compensating  brushes  were  not 
applied,  the  motor  would  then 
operate  as  a  plain  squirrel-cage 
induction  motor.  By  adding  the 
brushes  the  motor  can  be  com- 
pensated as  before.  It  is  evident 


that    considerable  of   the  current 
passing    into    the    compensating 


FIG.  50. — Armature  of    Heyland  Induc- 
tion Motor.' 


brushes   will "  leak  "  through  the 

resistors  connecting    the    commutator  bars.      It  is  therefore  essential 

that  these  resistors  have  sufficient  resistance,  or  an  excessive  loss  in 


SS 


THE  INDUCTION  MOTOR 


these  end  connections  will  result.     An  armature  of  this  type  is  shown  in 

Fig-  5°- 

The  action  of  the  commutator  in  correcting  the  current  of  line  fre- 
quency into  a  current  equivalent  to  one  of  the  frequency  of  slip,  is  very 
interesting,  and  in  view  of  the  fact  that  the  various  authors  who  have 
written  on  this  subject  do  not  agree  regarding  the  effect  of  this  action,  it 
is  perhaps  worthy  of  further  study. 

In  the  first  place,  it  is  self-evident  that  all  of  the  coils  between  two 
brushes  have  the  same  current  passing  through  them.  The  current  in 
the  section  between  any  two  brushes  will,  however,  in  general  be  different 
from  that  in  an  adjacent  section.  Moreover,  since  the  currents  in  two 
adjacent  sections  unite  to  form  the  current  in  the  brush,  these  currents 


FIG.  51. — Currents  in  Armature  Coils  of  a  Three-phase,  Commutator  Type,  Induc- 
tion Motor. 

must  be  sinusoidal  if  the  current  in  the  brush  is  sinusoidal.  That  this 
latter  is  strictly  true,  could  hardly  he  mantained,  even  though  the 
applied  e.m.f.  is  harmonic,  since  a  number  of  factors  will  intervene  to 
distort  the  current  wave  to  a  greater  or  less  extent.  In  general,  however, 
we  may  take  this  current  as  sinusoidal  without  serious  error. 

In  Fig.  51  we  have  drawn  three  sinusoids,  differing  in  phase  from  one 
another  by  120  degrees.  These  may  be  taken  as  being  the  three  currents 
in  three  adjacent  sections  of  a  rotor  fed  with  three-phase  current  by 
means  of  three  brushes  spaced  120  electrical  degrees  apart.  If  the 
rotor  is  at  rest,  these  curves  represent  the  currents  in  the  individual  coils 
of  the  rotor.  If  the  rotor  is  in  motion,  they  no  longer  represent  the  cur- 
rents in  any  particular  coil,  but  they  still  show  the  current  in  the  section 
of  the  rotor  between  the  brushes  corresponding  to  the  particular  curve. 

The  current  in  each  coil  is  determined  by  the  fact  that  as  long  as  the 
coil  in  question  is  between  two  brushes,  the  current  in  it  will  be  sinusoidal. 


VARIABLE  SPEED  INDUCTION  MOTORS  89 

As  the  coil  passes  under  the  brush  and  is  transferred  to  the  series  of  coils 
on  the  other  side,  the  value  of  the  current  in  it  will  be  suddenly  changed 
to  that  value  of  current  existing  in  the  other  series.  A  coil  which  hap- 
pens to  come  to  the  brush  at  the  time  when  the  current  is  the  same  in  the 
two  series  of  coils,  will  of  course  undergo  no  change  at  the  time.  This 
will  be  true  of  a  coil  arriving  at  a  brush  at  the  time  when  the  current  has 
the  value  "  a  "  in  Fig.  51.  The  curve  of  current  for  this  coil  will  be 
shown  by  the  line  abc.  A  coil  located  20  electrical  degrees  from  the  one 
first  considered  would  reach  the  brush  at  the  time  when  the  current  had 
the  value  a'.  The  current  in  it  would  assume  successively  the  values 
represented  by  the  curve  from  a'  to  b'.  As  it  passed  from  under  the 
brush,  the  current  would  change  suddenly  from  the  value  b'  to  c'.  The 
same  cycle  would  then  be  repeated. 

Likewise,  a  coil  located  20  electrical  degrees  from  the  second  coil 
would  have  current  in  it  passing  through  the  values  represented  by  the 
portion  jof  the  curve  marked  a"  b"  c".  The  greatest  variation  of  current 
while  the  coil  was  under  the  brush  would  occur  in  the  coil  situated  60 
electrical  degrees  from  the  first  coil  considered.  The  curve  of  current 
in  this  case  is  given  by  a'"  b'"  c'". 

If  instead  of  using  three  brushes  on  the  commutator  and  supplying 
three-phase  current,  four  brushes  had  been  used  with  two-phase  current, 
the  variation  of  current  in  the  coils  would  have  been  less.  This  appar- 
ent anomaly  is  explained  by  the  fact  that  in  this  case  we  should  have 
really  had  in  the  armature  a  four-phase  current.  In  the  limit,  with  an 
infinite  number  of  brushes  supplied  with  current  of  a  corresponding 
number  of  phases  we  should  have  no  change  in  current^  as  the  coil  passes 
under  the  brush,  or  in  other  words,  each  coil  would  carry  a  direct  current 
of  constant  strength. 

The  magnitude  of  the  e.m.f.  that  must  be  applied  to  the  brushes  to 
establish  the  current  through  the  winding,  depends  upon  the  shape  of  the 
magnetic  field  set  up  by  these  currents.  If  this  latter  were  harmonic  in 
space  variation  and  rotated  at  constant  speed  there  would  obviously  be 
no  e.m.f.  at  the  rotor  brushes  at  synchronous  speed,  due  to  the  cutting  of 
this  flux  by  the  conductors.  This  is  the  case  when  we  have  an  infinite 
number  of  brushes  supplied  from  an  infinite  number  of  phases. 

In  the  ordinary  case  of  three  or  four  phases,  we  do  not  have  this  con- 
dition. Instead  of  each  conductor  carrying  a  current  proportional  to 
the  sine  of  the  electrical  angle  of  jthe  conductor  from  a  fixed  point  of 
reference,  we  have  a  band  of  current  whose  width  is  1 20  or  90  electrical 
degrees,  and  the  strength  of  the  current  is  the  same  in  all  of  the  conduc- 


90  THE  INDUCTION  MOTOR 

tors  of  the  band.  If  as  is  sometimes  the  case,  the  direct-current  winding 
is  placed  on  the  armature  in  addition  to  a  squirrel-cage  winding,  or  if  the 
commutator  bars  are  connected  together  by  resistors  as  shown  in  Fig.  50, 
there  will  be  a  powerful  action  due  to  the  squirrel-cage  winding,  tending 
to  cause  the  flux  to  become  harmonic.  This  was  fully  explained  in  con- 
nection with  the  elementary  action  of  the  induction  motor. 

Even  though  such  a  winding  is  not  present,  there  will  be  a  tendency 
to  such  action  due  to  the  stator  winding.  If  the  flux  is  harmonic  and 
rotates  uniformly,  the  action  upon  the  stator  conductors  will  be  to  set  up 
a  harmonic  e.m.f.  at  the  terminals.  This  e.m.f.  is  nearly  equal  and 
opposite  to  the  applied  harmonic  e.m.f.  Any  other  variation  of  the  flux 
would,  however,  set  up  harmonics  of  e.m.f.  of  higher  frequency  in  the 
stator  winding.  These  higher  harmonics  would  be  short  circuited 
through  the  resistance  and  impedance  of  the  stator  winding  line  and  the 
generator  supplying  the  current.  The  resistance  and  impedance  of 
these  would  be  small,  particularly  in  case  the  motor  were  operated  from 
a  generator  of  large  size  and  through  a  short  transmission  line.  These 
short-circuited  harmonics  would  therefore  set  up  currents,  which  currents 
would  be  in  such  a  direction  as  to  tend  to  destroy  the  flux  which  caused 
them.  It  is  apparent,  then,  that  even  in  this  case,  there  would  be  a  con- 
siderable tendency  to  force  the  flux  to  become  harmonic.  This  action  is 
similar  to,  but  somewhat  weaker  than,  the  action  of  a  phase-wound  rotor 
upon  the  ordinary  induction  motor. 

The  above  refers  of  course  to  the  main  flux  of  the  motor,  i.e.,  that 
through  both  the  stator  and  the  rotor.  In  addition,  there  is  a  leakage 
flux,  and  the  reactance  due  to  this  would  be  approximately  the  same 
whether  the  rotor  were  at  rest  or  in  motion. 

At  any  other  than  synchronous  speed,  the  curve  of  current  in  any 
conductor  would  change  through  the  various  shapes  shown.  In  this 
case,  since  the  flux  and  the  coils  are  traveling  at  different  speeds,  there 
will  be  an  e.m.f.  due  to  the  coils  cutting  through  the  flux.  This  will  be 
in  such  a  direction  that  the  applied  e.m.f.  must  be  increased  for  speeds 
below  synchronism,  and  decreased  and  finally  reversed  for  those  above. 
From  this  it  will  be  apparent,  as  was  previously  pointed  out,  that  in  order 
to  secure  power-factor  compensation  for  an  adjustable  speed  motor  of  this 
type,  it  would  be  necessary  to  vary  the  e.m.f.  applied  to  the  compensating 
brushes  at  the  same  time  that  the  speed  was  varied. 

It  will  also  be  apparent  that  for  this  particular  type  of  motor,  a  two- 
phase  current  supply  to  the  rotor  would  have  some  advantage  over  a 
three-phase  supply.  It  might  be  that  in  certain  cases  this  would  be  of 


VARIABLE  SPEED  INDUCTION  MOTORS  91 

sufficient  importance  to  justify  the  transformation  of  a  three-phase  supply 
to  the  stator,  into  a  two-phase  current  for  the  rotor.  This  could,  of  course, 
be  readily  done  by  means  of  Scott  transformers  or  auto-transformers. 
Even  better  results  would  be  secured  by  using  six  sets  of  brushes  per 
pair  of  poles,  and  operating  the  rotor  six  phase. 

It  will  also  be  apparent  that  besides  the  method  explained  of  applying 
the  power  current  and  the  compensating  currents  to  separate  sets  of 
brushes,  the  same  effect  might  be  produced  by  applying  current  to  one 
set  of  brushes,  situated  in  an  intermediate  position.  The  rotating  band 
of  current  set  up  in  the  rotor  may  be  considered  as  resolved  into  two 
bands  rotating  in  the  same  direction.  One  of  these  would  lie  in  such  a 
position  as  to  coincide  with  the  rotating  band  of  flux,  and  consequently 
would  be  in  position  to  produce  the  torque  of  the  motor.  The  other 
would  be  situated  90  electrical  degrees  from  the  first,  and  would  produce 
no  torque,  but  would  be  in  the  proper  position  to  supply  the  magneto- 
motive force  necessary  to  force  the  flux  across  the  gap.  This  latter  com- 
ponent would  be  nearly  constant,  and  consequently  it  would  be  necessary 
to  shift  the  brushes  whenever  the  strength  of  the  rotor  curernt  was 
changed.  The  same  result  may  obviously  be  obtained  by  varying  the 
phase  of  the  e.m.f .  supplied  and  keeping  the  brushes  in  a  fixed  position. 
This  variation  in  the  phase  of  the  applied  e.m.f.  could  be  accomplished 
in  various  ways.  One  of  the  most  obvious  of  these  would  be  to  supply 
each  of  the  phases  of  rotor  current  by  means  of  two  transformers  con- 
nected in  series,  and  arranged  with  their  e.m.fs.  at  right  angles.  By 
varying  or  reversing  the  components  of  the  e.m.f.  any  phase  desired 
could  be  supplied. 

To  determine  the  effect  of  the  rotation  in  reducing  the  reactance  of  a 
rotor,  a  test  was  made  upon  a  small  wave-wound  armature.  This  was 
supplied  with  two  sets  of  brushes,  located  90  electrical  degrees  apart 
The  rotor  was  mounted  in  the  stator  of  an  induction  motor.  This  stator 
was  provided  with  a  single-phase  winding.  The  iron  was,  however, 
slotted  uniformly  all  around  the  circumference,  and  as  the  stator  winding 
was  not  used  in  the  experiment,  the  fact  that  it  was  single-phase  made 
no  difference.  Two-phase  current  at  a  voltage  of  about  22  volts  and  a 
frequency  of  45  cycles  was  supplied  to  the  rotor.  The  armature  was 
rotated  at  various  speeds,  both  in  the  direction  of  rotation  of  the  rotating 
magnetic  field  and  in  the  opposite  direction,  and  the  volts  and  amperes 
in  both  phases  measured.  From  these  data  the  impedance  per  phase  at 
various  speeds  can  be  readily  computed.  The  results  are  plotted  in  the 
curves  of  Fig.  52,  the  ordinates  being  values  of  the  impedance,  and  the 


92 


THE  INDUCTION  MOTOR 


abscissas,  the  speed  of  the  motor  in  per  cent  of  synchronism.  These 
two  curves  are  marked  "  impedance,"  and  the  one  at  the  right  is  plotted 
to  a  scale  ten  times  as  large  as  the  other,  to  show  the  rapid  change  near 
the  point  of  synchronous  speed.  As  will  be  seen,  the  impedance 
decreases  very  rapidly  as  synchronism  is  approached,  and  increases 
again  beyond  this  point.  The  negative  values  of  speed  correspond  to 
rotation  of  the  motor  in  opposition  to  the  rotation  of  the  magnetic  field. 
It  will  be  noted  that  the  point  of  minimum  impedance  is  attained  at 
a  speed  of  approximately  1 10  per  cent  of  synchronous  speed.  To  under- 
stand this,  we  must  remember  that  the  e.m.f.  generated  in  the  conduc- 
tors of  the  armature,  due  to  the  cutting  of  the  flux,  is  in  a  direction  to 


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Backwai-d  Rotation  Foreward  Rotation 

FIG.  52.— Impedance  and  Reactance  Curves. 

oppose  the  applied  e.m.f.  below  synchronism,  and  in  a  direction  to  help 
it  above  that  speed.  It  is  probable  that  at  synchronism,  the  reactance 
is  nearly  zero  and  the  principal  opposition  to  the  current  is  the  ohmic 
resistance  of  the  winding.  Moreover,  the  e.m.f.  required  to  overcome 
the  reactance  is  at  right  angles  to  that  over  the  resistance.  The  influ- 
ence of  the  reactance  is  therefore  slight  at  speeds  near  synchronism,  and 
since  above  this  point,  the  generated  e.m.f:  is  in  the  proper  direction  to 
help  the  applied  e.m.f.,  the  impedance  continues  to  decrease.  Since, 
however,  the  reactance  increases  almost  in  proportion  to  the  departure 
of  the  speed  from  synchronism,  while  the  total  e.m.f.,  i.e.,  the  sum  of  the 
applied  and  the  counter  e.m.f.  is  in  proportion  to  the  increase  of  the 
speed  above  zero,  the  increased  reactance  soon  shows  its  influence  and 
the  impedance  again  increases. 

An  attempt  was  also  made  to  separate  the  effect  of  resistance  and 
reactance  in  the  curves.     This  was  rendered  difficult  on  account  of  the 


VARIABLE  SPEED  INDUCTION  MOTORS  93 

fact  that  the  resistance  of  a  rotor  and  brushes  is  a  function  not  only  of 
the  current,  being  very  much  less  with  large  currents  than  with  small,  but 
it  is  also  considerably  influenced  by  the  speed  of  the  machine.  This  is  on 
account  of  the  varying  contact  of  the  brushes,  due  to  vibration  and  other 
causes.  The  resistance  was  measured  with  direct  current,  the  rotor 
being  in  motion  at  about  synchronous  speed.  The  results  were  plotted 
in  the  form  of  a  curve,  and  from  this,  the  resistance  drop  corresponding 
to  any  current  could  be  readily  found.  This  e.m.f.  was  subtracted  vec- 
torially  from  the  total  applied  e.m.f.  The  difference  divided  by  the 
current  gave  the  reactance.  These  results  are  plotted  in  the  curve 
marked  "reactance."  It  of  course  differs  little  from  the  impedance 
curve  for  large  values  of  the  impedance,  but  falls  very  much  below  it  for 
the  points  near  synchronism.  Indeed,  judging  from  the  results,  the 
reactance  was,  as  nearly  as  could  be  determined,  zero,  for  speeds  from 
about  95  per  cent  to  105  per  cent  of  synchronism.  It  was  not  possible 
at  the  time  to  do  the  work  with  sufficient  accuracy  to  determine  these 
values  more  definitely.  In  any  event  it  is  certain  that  the  reactance  at 
synchronous  speed  was  so  nearly  zero  that  it  would  have  been  necessary 
to  adopt  special  precautions  to  discover  its  existence. 

A  similar  test  was  made,  the  rotor  being  supplied  with  single-phase 
current.  Practically  no  change  in  the  impedance  could  be  discovered 
as  the  speed  was  changed.  The  constant  value  of  the  impedance  was 
7.3  ohms.  That  there  is  no  change  in  this  case  is  due  to  the  fact  that 
with  the  single-phase  current  supplied  only  to  the  rotor,  we  did  not  have 
a  rotating  magnetic  field.  The  counter  e.m.f.  did  not  therefore,  decrease 
in  the  manner  described  in  connection  with  the  two-phase  current.  A 
single-phase  motor  in  operation  would  in  general  have  nearly  a  true 
rotating  magnetic  field  and  the  apparent  impedance  would  decrease  the 
same  as  in  a  polyphase  motor. 

COMMUTATION 

When  the  rotor  is  operating  at  synchronism,  the  rotor  conductors  are 
moving  at  the  same  velocity  as  the  stator  flux.  There  is  consequently 
no  e.m.f.  generated  in  them  and  no  current  produced  in  the  turns  short 
circuited  under  the  brushes.  There  is,  of  course,  a  small  reactance  volt- 
age due  to  the  sudden  reversal  of  the  current  in  the  coil,  as  in  the  case  of  a 
direct-current  machine,  but  by  suitable  design  this  can  readily  be  kept 
small  enough  so  that  there  will  be  no  sparking.  About  the  only  feature 
which  renders  the  commutation  more  difficult  than  that  of  a  direct-cur- 
rent machine  is  the  fact  that,  with  the  same  virtual  value  of  direct  or 


91  THE  INDUCTION  MOTOR 

alternating  current,  the  maximum  of  the  wave  in  the  case  of  the  alter- 
nating current  may  have  to  be  commutated,  and  this  is  41  per  cent 
greater  than  the  virtual  value.  It  would  therefore  be  necessary  to  keep 
the  inductance  of  the  coils  somewhat  lower  than  in  the  case  of  a  direct- 
current  machine. 

At  any  other  than  synchronous  speed,  however,  the  conditions  would 
not  be  so  favorable.  Consider,  for  example,  the  rotor  at  standstill.  Prac- 
tically the  same  flux  would  cut  both  the  rotor  and  the  stator  conductors 
and  consequently  there  would  be  generated  in  each  turn  of  the  rotor  prac- 
tically the  same  e.m.f.  as  in  each  turn  of  the  stator.  Since  we  cannot 
well  have  less  than  one  turn  of  the  rotor  conductor  between  each  pair  of 
commutator  bars,  we  have  short  circuited  at  the  brushes  an  e.m.f.  equal 
to  the  e.m.f.  per  stator  turn.  This  e.m.f.  will  set  up  a  large  current  in 
the  short-circuited  rotor  conductors.  If  there  were  no  magnetic  leak- 
age, the  short-circuit  current  would  assume  such  a  value  that  the  ampere- 
turns  in  the  short-circuited  rotor  conductors  would  be  equal  to  the 
ampere-turns  of  all  the  stator  conductors.  Since,  however,  the  two  wind- 
ings are  on  separate  cores  and  since  the  resistance  of  the  brush  contacts 
and  the  rotor  conductors  is  considerable,  the  current  does  not  rise  to 
nearly  this  value,  but  would,  if  not  prevented  in  any  way,  rise  to  several 
times  its  normal  full-load  value. 

Precisely  the  same  difficulty  is  met  with  in  the  design  of  single-phase 
railway  motors.  The  solution  of  the  difficulty  is  usually  accomplished 
in  one  of  two  ways :  Either  the  field  flux  is  kept  very  weak,  so  that  the 
generated  e.m.f.  is  small,  or  else  resistance  leads  connecting  the  winding 
to  the  commutator  are  used.  These,  it  will  be  readily  seen,  are  in  circuit 
only  with  the  current  from  the  commutator  bars  under  the  brush  to  the 
rotor  winding,  and  are  not  in  circuit  with  the  current  from  coil  to  coil. 
The  PR  loss  in  them  is  therefore  small,  and  their  use  does  not  materially 
lower  the  efficiency. 

We  have  considered  the  commutator-type  motor  with  the  rotor  and 
stator  connected  in  parallel  to  the  supply  circuit.  There  is,  however,  no 
reason  why  they  should  not  be  connected  in  series  to  the  line.  The 
motor  would  then  have  series  characteristics  instead  of  shunt  characteris- 
tics as  in  the  other  case,  i.e.,  it  would  slow  down  greatly  under  load  and 
speed  up  greatly  when  the  load  was  removed.  It  would  in  fact  have 
essentially  the  same  characteristics  as  the  single-phase,  commutator  type 
of  motor,  except  that  the  torque  would  be  constant  during  the  revolution, 
instead  of  intermittent.  The  motor  could  probably  be  built  somewhat 
lighter  for  the  same  output,  but  this  advantage  would  be  at  least  par- 


VARIABLE  SPEED  INDUCTION  MOTORS  95 

tially  offset,  especially  for  railway  work,  by  the  necessity  of  supplying 
three  conductors  instead  of  two.  It  should  also  be  pointed  out  that  if 
a  motor  of  this  type  were  used  for  railway  work,  it  would  be  possible  to 
arrange  the  circuits  so  that  it  could  be  used  on  direct  current  as  well  as 
on  alternating.  This  would  of  course  in  general  be  essential  in  the  case 
of  interurban  railroads. 

This  type  of  motor  has  not  as  yet  found  extensive  practical  applica- 
tion. Its  possibilities  are,  however,  certainly  great,  and  it  is  entirely  pos- 
sible that  the  demand  for  an  adjustable-speed  induction  motor  may 
bring  it  into  general  use.  A  single-phase  motor,  operating  upon  the 
same  general  principles  and  having  a  compensating  winding,  is  now 
on  the  market.  This  is  fully  treated  in  Chapter  XIII. 


CHAPTER   VII 
RELATIONS  OF  FLUX  E.M.F.  AND   CURRENT 

WE  have  shown  in  Chapter  I  that  in  the  case  of  an  induction  motor, 
particularly  if  of  the  squirrel-cage  type,  when  operating  at  or  near  syn- 
chronous speed,  the  distribution  of  flux  in  space  is  very  nearly  sinusoidal, 
provided  the  applied  e.m.f.  is  also  harmonic.  This  is,  moreover,  true 
irrespective  of  the  type  of  winding  employed,  and  irrespective  of  whether 
or  not  full-pitch  or  fractional-pitch  coils  are  used.  Various  circum- 
stances may  modify  somewhat  this  distribution  of  the  flux.  As  far  as 
the  stator  current  is  concerned,  its  tendency  will  always  be  to  give  a  more 
or  less  step-like  distribution  as  shown  in  Figs.  8  and  9.  The  rotor  of  a 
squirrel-cage  motor  has  a  powerful  damping  effect  upon  any  variation  of 
the  flux  and  consequently  tends  to  force  the  flux  wave  to  retain  some 
given  shape.  The  only  distribution  which  will  always  give  a  sine  wave  of 
counter  e.m.f.  is  the  sine  wave,  consequently  it  is  the  one  we  should  con- 
sider in  any  general  discussion. 

If  the  squirrel-cage  winding  is  of  high  resistance  so  as  to  give  large 
starting  torque,  it  is  evident  that  the  damping  action  spoken  of  will  be 
much  weaker,  and  the  wave  of  flux  may  depart  somewhat  from  the  sine 
shape.  It  is  also  evident  that  a  rotor  of  the  slip-ring  type  will  have  a 
much  weaker  damping  effect  than  a  squirrel-cage  rotor.  This  is  true 
since  the  corrective  currents  are  not  free  to  follow  the  exact  path  required 
to  produce  the  correction,  but  are  forced  to  follow  through  the  windings 
in  the  order  of  their  connection.  These  corrective  currents  themselves 
will  tend  to  set  up  variations  in  the  flux  and  will  require  currents  in  the 
other  windings  to  reduce  this  variation.  The  correction  is  consequently 
much  less  complete  in  the  case  of  a  slip-ring  motor. 

If  considerable  resistance  is  inserted  in  the  rotor  circuit  of  a  slip-ring 
machine,  the  corrective  effect  is  still  less,  and  obviously  on  open  circuit 
the  effect  will  be  zero. 

The  number  of  stator  slots  also  has  a  great  influence  upon  the  flux 
distribution.  If  the  number  of  slots  be  made  very  great,  the  steps  in  the 

96 


RELATIONS  OF  FLUX  E.M.F.  AND  CURRENT  97 

flux  curve  will  be  small,  and  in  the  limit,  with  an  infinite  number  of  slots, 
would  blend  into  a  smooth  curve.  To  recapitulate,  then,  we  shall  have 
the  nearest  to  a  sine  distribution  of  the  flux  when  the  number  of  stator 
slots  is  great,  the  rotor  winding  is  of  the  squrirel-cage  type,  with  many 
bars,  and  is  of  low  resistance.  On  the  other  hand,  the  wave  will  depart 
most  widely  from  the  sine  shape  when  the  stator  winding  is  in  few  slots, 
and  the  rotor  is  of  the  slip-ring  type  and  is  on  open  circuit. 

Assuming,  then,  that  the  wave  of  flux  is  distributed  in  space  accord- 
ing to  a  sine  law,  the  e.m.f.  generated  in  N  conductors  connected  in 
series,  and  supposed  to  be  arranged  in  one  slot  per  phase  per  pole  will  be 
given  by  E  —  2.22®  Nf+  io8,  in  which  $  is  the  total  flux  per  pole,  and/ is 
the  frequency.  This  is  the  usual  equation  of  a  transformer  or  alterna- 
tor. In  practice,  however,  the  conductors,  except  in  the  case  of  exceed- 
ingly small  machines,  are  never  disposed  in  only  one  slot  per  phase  per 
pole.  As  a  consequence  the  e.m.f s.  generated  in  the  various  coils  of  the 


FIG.  53. — Vector  Diagram  of  E.M.Fs. 

same  winding  are  not  in  the  same  phase,  but  differ  by  an  angle  equal  to 
360  divided  by  the  number  of  slots  corresponding  to  two  poles.  Thus 
in  a  three-phase  four-pole  motor,  having  36  slots,  the  e.m.f.  in  adjacent 
coils  will  differ  by  20  degrees. 

Such  a  motor  would  probably  be  wound  with  thirty-six  coils,  or  three 
coils  per  phase  per  pole.  The  e.m.f.  generated  by  each  coil  might  be 
computed  and  laid  off  as  in  Fig.  53,  the  angle  between  successive  vectors 
being  180 —  20  =  160  degrees.  Thus  AB  represents  the  e.m.f.  of  the  first 
coil,  EC  that  of  the  second,  and  CD  that  of  the  third.  The  line  AD 
represents  both  in  magnitude  and  phase  the  value  of  the  resultant,  and 
we  could  readily  compute  its  value  by  the  ordinary  methods  of  trigonome- 
try. If,  however,  we  should  adopt  this  plan,  it  would  be  necessary  to 
compute  a  separate  value  for  each  number  of  slots  per  pole.  It  will  be 
readily  seen  that  the  different  values  will  not  be  greatly  different,  and  it 
is  apparent  that  we  can  compute  one  value  that  will  be  near  enough  for 
all  practical  purposes.  This  is  especially  true,  since  it  is  in  general 
unnecessary  to  compute  our  values  to  any  very  great  degree  of  precision, 
as  many  of  our  assumptions  are  necessarily  not  very  exact. 

The  simplest  assumption  to  make  is  that  we  have  an  infinite  number 


98 


THE  INDUCTION  MOTOR 


of  coils  per  phase  per  pole.     The  vector  diagram  would  then  be  as  repre- 
sented in  Fig.  54.     With  an  infinite  number  of  slots,  the  e.m.f.  due  to  the 

coils    in    each    slot    will    be    in- 
_____  -  -------  finitesimally    small    compared    to 

^^__  __    ^""-^       the    whole    e.m.f.    and    will    be 

represented  merely  by  a  dot  on 
the  segment  of  the  circle.  The 
resultant  e.m.f.  is  the  straight 
line  AB.  What  we  have  to  obtain 
is  the  ratio  of  the  straight  line 
to  the  segment  of  the  circle. 
This  ratio  will  give  us  the  ratio 
in  which  the  e.m.f.  is  reduced 
on  account  of  the  fact  that  the 
coils  are  distributed,  instead  of 
being  concentrated  in  one  slot. 
This  ratio  for  the  three-phase 


FIG.  54. — E.M.F.  Diagram,  Infinite 
Number  of  Coils. 


motor  is  evidently 


In  the  case  of  the  two-phase  winding,  the  angle  spanned  by  the  coils 
is  90  degrees  instead  of  60  degrees.     Consequently  the  ratio  is 


2  sin  45°     2  XV  2 


7T/2 


=  0.90. 


The  apparent  discrepancy  betweed  the  angle  60  degrees  in  a  three- 
phase  and  90  degrees  in  a  two-phase  winding  is  reconciled,  when 
we  consider  that  all  three-phase  windings,  except  in  the  case  of  the 
synchronous  converter,  are  really  six-phase  windings,  reconverted  so 
as  to  give  three  phases.  The  true  three-phase  connection  would  give 
a  reduction  factor  of  0.827.  The  factor  for  a  six-phase  winding  is, 
as  shown,  0.953.  Hence  the  six-phase  winding  will  give  a  larger  out- 
put, and  is  therefore  preferable. 


SHORT-PITCH  WINDINGS 

Most  modern  induction  motors  are  wound  with  short-pitch  or  frac- 
tional-pitch windings,  that  is,  the  coils  instead  of  spanning  a  complete 
pole  pitch  or  180  degrees,  fall  short  of  this  by  one  or  more  slots.  The 


RELATIONS  OF  FLUX  E.M.F.  AND  CURRENT  99 

reasons  for  adopting  this  type  of  winding  are  fully  discussed  elsewhere. 
On  account  of  the  fractional  pitch,  the  e.m.fs.  of  the  two  sides  of  a  coil  are 
not  in  the  same  phase  and  consequently  the  counter  e.m.f .  due  to  a  given 
flux  is  less,  or  conversely,  the  flux  to  produce  a  given  counter  e.m.f.  must 
be  greater.  The  ratio  to  apply  to  the  whole  winding  is  obviously  the 


FIG.  55.—  Vector  Diagram,  Fractional  Pitch  Coils. 

same  as  the  ratio  for  one  coil.  Thus  in  Fig.  55  this  ratio  is  equal  to  the 
length  of  the  line  AC  divided  by  the  algebraic  sum  of  the  lines  AB  and 
EC.  If  we  make  AB  and  BC  each  equal  to  one,  the  ratio  becomes 

sin  —  ,  where  f  is  the  electrical  angle  between  the  two  sides  of  the  coil. 

There  is  one  other  respect  in  which  the  induction  motor  differs  from 
the  transformer.  The  flux  along  the  gap  is  not  distributed  in  a  uniform 
manner,  but  follows  a  sine  law.  We  are  usually  most  concerned  with 
the  maximum  value  of  this  flux,  rather  than  with  its  average  value.  In 
any  sine  wave  the  ratio  of  the  average  value  to  the  maximum  value  is 

—  .  If  then  we  designate  by  (Bm  the  maximum  value  of  the  flux  density 
and  by  <•/>  the  total  flux,  we  have, 

(f>=-C&mA. 

7T 

where  A  is  the  area  of  the  pole. 

Returning  now  to  our  transformer  equation 


and  substituting  for  0  the  value  obtained  above  and  multiplying  by 
0.953  in  the  case  of  a  three-phase  winding  or  by  0.90  in  the  case  of  a  two- 

phase,  and  by  sin  —  to  allow  for  short-pitch  windings,  we  obtain 

E  (three-phase)  =  ^JA  (Bm/7V    or    E  (two-phase)  =^j^-A®mfN. 
For  our  purposes  the  value  of  (Bm  is  usually  the  one  to  be  obtained, 


100  THE  INDUCTION  MOTOR 

and  for  this  purpose  the  equations  can  be  more  conveniently  written  in 
the  form 

_7.42£io7 
(Bw~    ANf 
for  three-phase  windings;   or, 


for  two-phase  windings.      If  fractional   pitch-windings   are  employed 
the  equations  become  respectively, 

7.42.Eio7 

and  (Bm  =  - 


A  Nf  sin  f/z  ANfsmr/2 

These  equations  are  the  same  whether  (Bm  and  A  are  expressed  in 
centimeter  or  inch  units. 


THE  MAGNETIZING  CURRENT 

The  determination  of  the  magnetizing  current  is  one  of  the  most 
difficult  problems  with  which  we  have  to  deal.  This  is  due  not  so  much 
to  any  complexity  in  the  equations  involved,  as  it  is  to  the  uncertainty  of 
the  assumptions  involved.  The  problem  can  be  attacked  in  a  variety  of 
ways,  and  in  general  the  results  will  differ  somewhat.  However,  the 
differences  are  not  great,  and  in  no  case  even  approach  the  uncertainty 
introduced  by  the  varying  length  of  the  air-gap,  which  is  inevitable  in 
practice. 

One  method  of  attack  is  to  assume  sine  waves  of  current  in  the  stator, 
and  no  rotor  current.  The  waves  of  flux  will  consequently  be  stepped 
as  shown  in  Figs.  8  and  9,  and  will  vary  from  point  to  point.  This 
assumption  may  be  approximately  true  in  the  case  of  an  induction  motor 
with  wound  rotor,  when  on  open  circuit,  but  it  is  very  far  from  true  in 
the  case  of  the  squirrel-cage  machine  or  in  the  case  of  the  wound-rotor 
machine  when  operating  normally  with  secondary  short  circuited.  The 
reason  for  this  fact  has  been  pointed  out  in  Chapter  I. 

The  method  which  on  the  whole  seems  the  simplest,  is  to  assume  that 
the  wave  oiflux  is  a  sine  wave.  It  follows  that  the  magnetizing  current 
will  not  follow  a  sine  law,  but  we  can  calculate  the  value  of  the  equivalent 
sine  wave. 

Fig.  56  represents  a  section  of  the  stator  and  rotor  of  an  induction 
motor.  The  winding  is  supposed  to  be  full  pitch.  The  small  circles 
represent  currents  coming  toward  the  observer,  the  small  crosses  cur- 


RELATIONS  OF  FLUX  E.M.F.  AND  CURRENT  101 

rents  in  the  opposite  direction.  The  point  of  maximum  flux  will  be  at 
the  position  shown.  All  the  currents  at  the  left,  as  well  as  all  those  at 
the  right,  tend  to  produce  flux  in  an  upward  direction,  i.e.,  from  the 
rotor  to  the  stator.  The  student  should  carefully  note  that  the  distance 
of  the  conductor  from  the  point  A  has  in  itself  nothing  to  do  with  the 
tendency  which  it  has  to  produce  flux  across  the  gap.  It  might  appear 
at  first  sight  that  the  nearer  conductors  would  exert  a  greater  effect  for 
the  same  current  than  those  farther  away.  That  this  is  not  so  is  due  to 
the  fact  that  although  the  straight  parts  of  the  conductors  further  away 
do  not  exert  so  great  an  action  as  do  those  nearer,  the  end  connections  of 
the  conductors  further  away  are  longer,  and  consequently  make  up  for 
the  lesser  effect  of  the  straight  portions.  This  is  equivalent  to  the  state- 


aaa     aabbblbbbc     c 

o  o   o    o   o   o    o  o|x    xxx    x    x    x   x   x 


FIG.  56. — Magnetizing  Currents  of  Induction  Motor. 

ment  that  the  magnetic  force  at  the  center  of  a  circular  wire  carrying  a 
current  is  independent  of  the  diameter  of  the  circle. 

For  simplicity  let  us  assume  that  the  number  of  phases  is  very  great, 
and  that  each  carries  a  sinusoidal  current.  Let  the  number  of  phases  be 
y,  and  let  the  number  of  conductors  per  phase  per  pole  be  A^.  Then 
the  number  of  turns  acting  on  the  iron  at  the  center  of  the  pole  is  yNi  -=-  2. 
Let  the  maximum  current  in  any  conductor  be  Im.  The  average  value 

of  the  current  is  obtained  by  multiplying  by  — ;  or, 

K 

Average  current  =7mX — 

7T 

Also  let  im  be  the  virtual  value,  that  is,  the  value  measured  by  an  ammeter 
of  the  current  in  any  conductor,  then 


Average  current  =  Im  X  - = .*'»»• 

7T  7T 


102  THE  INDUCTION  MOTOR 

The  magnetomotive  force  is  equal  to  47?+- 10  times  the  current  and  times 
the  number  of  turns,  or 

AX  yN-i      /—.    2 
m.m.f.  = — .V  2im.—  =o.s66Niyim. 

10        2  71 

The  value  of  the  flux  per  square  centimeter  is  given  by  the  m.m.f. 
divided  by  the  length  of  path  in  the  air.  This  is,,  making  no  allowance 
for  the  part  of  the  path  through  the  iron.  Then 


Solving  this  for  im  we  get, 
or  in  inch  units, 


.  ^0.695  ($>md 

""  ' 


This  was  developed  on  the  assumption  of  a  very  great  number  of 
phases.  To  assume  that  the  same  formula  will  hold  for  a  small  number 
of  phases  as  two  or  three,  is  not  strictly  correct.  That  this  is  so  is  appar- 
ent from  the  fact  that  with  a  small  number  of  phases  the  current  in  the 
conductors  is  not  distributed  along  the  gap  in  accordance  with  a  sine  law. 
The  current  is  in  fact  distributed  in  a  small  number  of  bands,  as  shown 
in  Fig.  5.  The  current  in  one  edge  of  the  band  is  stronger  than  we  have 
assumed  in  developing  this  theory,  and  in  the  other  edge  it  is  weaker.  As 
we  have  previously  shown,  however,  the  distribution  of  the  flux  remains 
sinusoidal.  In  order  that  this  may  be  so,  it  is  necessary  that  currents 
circulate  in  the  rotor  bars  in  such  a  manner  as  to  offset  this  tendency  of 
the  stator  currents  to  give  a  stepped  distribution  of  the  flux.  Thus  in 
Fig.  56  the  letters  a,b  and  c  represent  the  conductors  belonging  to  the  a, 
the  b,  and  the  c  phase  respectively  of  a  three-phase  machine.  It  is 
evident  that  the  current  is  the  same  in  all  of  the  a  conductors,  and  like- 
wise in  all  of  the  b  and  the  c  conductors.  In  Fig.  57  let  us  consider  the 
instant  when  the  a  current  is  zero,  and  the  B  and  C  currents  are  equal 
and  opposite.  The  distribution  of  the  current  in  the  stator  is  as  indi- 
cated by  the  full  line  marked  stator  current.  The  flux,  however,  on 
account  of  the  rotor  reaction,  follows  a  sine  shape  as  shown.  To  produce 
this  flux  by  a  stator  current  alone  would  require  a  current  distribution 
as  shown  by  the  dotted  line.  It  is  evident  that  there  must  be  enough 
current  in  the  rotor  to  compensate  for  the  excess  or  deficiency  of  the 


RELATIONS  OF  FLUX  E.M.F.  AND  CURRENT 


103 


stator  current.  The  ordinates  of  this  current  are  as  shown  by  the 
dotted  line  marked  rotor  current.  Fig.  58  shows  a  similar  diagram 
for  the  case  when  the  currents  in  the  A  and  the  B  phase  are  equal,  and 
the  C  current  is  at  its  maximum  in  the  opposite  direction.  The  curve 


Equivalent  Magnetizing  Current 

Stator  Current Rotor  Current 

FIG.  57. — Distribution  of  Magnetizing  Currents  in  Stator  and  Rotor. 

of  rotor  current  distribution  is  much  more  distorted  in  this  case  than 
in  the  first  case.  Likewise  we  could  draw  diagrams  for  any  of  the 
values  of  the  stator  current.  It  will  be  observed  that  in  each  case  the 
rotor  current  distribution  curve  contains  a  fundamental  of  three  times 
the  primary  frequency. 


Equivalent  Magnetizing  Current 

Stator  Current Rotor  Current. 


FIG.  58. — Distribution  of  Magnetizing  Currents  in  Stator  and  Rotor. 

In  a  similar  manner  curves  could  be  drawn  representing  the  currents 
in  the  rotor  bars,  when  a  two-phase  stator  is  used.  We  can  even  extend 
the  method  to  the  case  of  a  single-phase  stator,  and  show  how  the  cur- 
rents in  the  rotor  act  to  maintain  an  approximately  sinusoidal  distribu- 
tion of  the  flux,  and  hence  a  uniform  rotating  magnetic  field.  This  case 
is  considered  more  in  detail  on  page  194,  and  the  curves  corresponding 
to  single-phase  operation  are  shown  in  Figs.  99  and  100. 


104  THE  INDUCTION  MOTOR 

It  must  be  remembered  in  considering  these  curves  of  rotor  and  stator 
currents,  that  they  do  not  represent  the  variation  with  time  of  the  current 
as  is  the  case  with  most  curves,  but  on  the- contrary  represent  the  space 
variation  of  the  current.  The  curves  as  shown  are  for  the  condition  of  no 
load  with  the  rotor  operating  as  synchronous  speed.  When  the  motor 
is  loaded,  there  is  added  to  the  stator  current  a  component  in  phase  with 
the  applied  e.m.f.  or  90  degrees  different  in  phase  from  the  magnetizing 
current  shown.  An  equal  and  opposite  current  is  produced  in  the  rotor, 
and  the  total  rotor  current  is  then  the  sum  of  the  added  component,  and 
the  corrective  current  shown  in  Figs.  50  and  51. 

It  is  evident  that  the  corrective  current  in  the  rotor  causes  some 
copper  loss  in  addition  to  that  due  to  the  stator  current.  However,  the 
loss  due  to  the  magnetizing  current  is  small  in  comparison  with  the  full- 
load  stator  loss,  and  since  the  corrective  current  in  the  rotor  is  consider- 
ably smaller  than  the  stator  magnetizing  current,  this  added  copper  loss 
may  well  be  neglected. 

It  will  be  evident  from  what  has  just  been  said  that  the  presence  of 
the  rotor  corrective  current  will  considerably  complicate  the  exact  com- 
putation of  the  magnetizing  current  of  a  commercial  induction  motor  of 
two  or  three  phases,  and  this  difficulty  will  be  considerably  increased  in 
the  case  of  a  motor  having  short-pitch  windings.  At  least  an  approxi- 
mate solution  can  be  obtained  in  the  following  manner;  In  a  three- 
phase  motor  having  say  three  slots  per  phase  per  pole,  it  will  be  evident 
that,  with  a  uniformly  revolving  sinusoidal  flux  the  e.m.f.  in  any  slot  will 
differ  20  degrees  from  that  generated  in, a  slot  on  either  side  of  it.  The 
currents,  however,  in  three  adjacent  slots  are  the  same,  since  all  of  these 
conductors  are  connected  in  series.  We  have  shown  that  for  the  best 
operation,  each  current  should  differ  from  its  neighbor  by  an  angle  corre- 
sponding to  the  angle  of  displacement  of  the  coils  along  the  core.  We 
may  consider  then  that  in  this  particular  case,  the  current  in  one  of  the 
three  slots  is  in  the  correct  phase  while  that  in  each  of  the  adjacent  slots 
differs  by  20  degrees  from  the  correct  phase.  The  ampere-turns  of 
each  of  these  should  therefore  be  multiplied  by  the  cosine  of  20  degrees. 
The  equivalent  magnetomotive  force  of  the  three  sets  of  coils  corre- 
sponding to  the  three  slots,  compared  to  that  which  we  should  have  if 
the  three  currents  were  in  the  theoretically  correct  phase  relation  is 

(i-f  2  cos  #)-=-3=96  per  cent. 

It  will  be  seen  that  this  process  of  analysis  is  identical  with  that 
employed  to  obtain  the  factor  by  which  the  e.m.f.  is  reduced  on  account 


RELATIONS  OF  FLUX  E.M.F.  AND  CURRENT  105 

of  the  fact  that  the  e.m.fs.  in  the  different  coils  of  a  winding  are  not  all  in 
the  same  phase.  We  may  conveniently  employ  the  method  used  in  that 
case,  and  consider  only  the  particular  instance  when  there  are  a  very 
great  number  of  conductors  per  phase  per  pole.  With  this  assumption, 
the  breadth  coefficient  is  found  to  be  0.952  for  a  three-phase  winding  and 
0.900  for  a  two-phase  winding.  It  will  be  seen  that  the  value  for  the 
limiting  three-phase  case  does  not  differ  materially  from  the  particular 
condition  chosen  of  three  slots  per  phase  per  pole. 

The  effect  of  short-pitch  winding  may  be  handled  in  a  similar  manner. 
In  the  ordinary  case  of  a  Iwo-layer  winding,  the  effect  of  the  short  pitch 
is  to  shift  the  one  layer  through  an  angle  equal  to  the  difference  between 
the  angle  spanned  by  the  coil  and  180  degrees.  The  magnetomotive 
force  of  each  layer  would  then  differ  by  half  this  angle  from  the  position 
corresponding  to  the  greatest  effect.  The  correction  factor  would  then 
be  the  cosine  of  half  of  this  angle  or  the  sine  of  half  the  angular  pitch  of 
the  coil.  This  is  again  identical  with  the  correction  factor  found  in  con- 
nection with  the  calculation  of  the  e.m.f. 

It  will  be  seen  that  the  use  of  each  of  these  factors  means  that  with 
a  given  maximum  value  of  the  flux  density,  we  must  multiply  the  e.m.f. 
by  a  certain  expression  to  correct  for  the  spread  of  the  coils,  and  by 
another  expression  to  allow  for  the  effect  of  short  pitch.  At  the  same 
time,  the  value  of  the  magnetizing  current  must  be  divided  by  the 
same  factors.  The  net  result  is  then  that  the  applied  voltage  is  reduced 
and  the  magnetizing  current  increased  by  the  same  factor.  The  value 
of  the  total  volt-amperes  is  therefore  the  same. 

This  result  was  pointed  out  by  Dr.  A.  S.  McAllister  in  his  book, 
"  Alternating  Current-  Motors."  As  is  there  shown,  this  result  might  be 
expected  from  the  fact  that,  since  the  actual  energy  expended  in  building 
up  the  magnetic  field  in  a  particular  place  (which  energy  is  again  restored 
to  the  circuit  when  the  field  at  the  point  falls  to  zero)  is  constant,  the 
apparent  watts,  i.e.,  the  volt-amperes,  should  also  be  constant.  Thus  in 
each  cubic  centimeter  of  air  in  which  the  magnetic  density  is  ($>  there  is 

(B2 

stored  energy  to  the  amount  of  —          —  -^  joules.     If  then,  the  shape  of 

8  J 


the  flux  wave  remains  sinusoidal,  the  actual  energy  of  the  stored  mag- 
netic field  remains  constant,  no  matter  what  the  type  of  the  winding  by 
which  it  is  produced,  and  we  should  expect  by  analogy  that  the  value  of 
the  apparent  watts  or  the  wattless  volt-amperes  would  remain  constant. 


106  THE  INDUCTION  MOTOR 

Dividing  by  the  number  of  phases  and  by  the  factors  indicated  for 
two-  and  three-phase,  and  also  dividing  by  sin  _  to  allow  for  fractional- 

pitch  winding  (the  factor  becomes  one,  for  full  pitch),  we  obtain  as  the 
final  expressions  for  the  magnetizing  current, 

d 

(Three-phase,  centimeter  units), 


I  sin  T- 


im=°-243(S*?d_     (Three-phase,  inch  units), 
A^sin^ 

2 

im=—  —  (Two-phase,  centimeter  units), 

A^sin^ 

(Two_phasej  inch  units) 
Nl  sin  r- 

To  test  the  above  theory,  use  was  made  of  a  small  squirrel-cage  induc- 
tion motor.  This  machine  was  wound  with  72  coils  in  the  same  number 
of  slots.  The  coils  were  connected  in  six  sets  of  12  coils  each.  The 
circuits  were  located  30  electrical  degrees  apart  on  the  core.  By  various 
groupings  of  the  coils,  the  machine  could  be  operated  as  a  single,  two-, 
three-  or  six-phase  machine.  To  insure  the  same  value  of  the  flux  in  all 
of  the  cases,  the  applied  voltage  was  varied  so  as  to  give  40,  50,  55  and 
60  volts  over  one  section  of  the  winding.  Thus  assuming  that  the  flux 
was  harmonic  or  at  least  did  not  change  with  the  change  in  connections, 
the  readings  were  taken  for  the  same  value  of  the  flux  in  each  case. 

Readings  were  made  as  above  indicated  with  the  machine  connected 
for  one,  two,  three  and  six  phases.  Having  adjusted  the  applied  voltage 
until  a  certain  voltage  was  indicated  across  one  of  the  coils,  readings  were 
taken  of  the  current  in  the  various  phases  and  of  the  voltage  across  the 
phases.  The  average  current  per  phase  was  then  multiplied  by  the  aver- 
age voltage  per  phase  and  by  the  number  of  phases.  The  result  is  the 
total  no-load  volt-amperes.  To  get  the  true  magnetizing  current  of 
the  motor,  it  would  have  been  necessary  to  subtract  vectorially  from  the 
no-load  current  the  power  component  of  the  no-load  current.  This  was 
not  done  hi  this  case,  as  it  would  have  changed  the  results  but  little.  The 
resultant  volt-amperes  with  the  different  connections  are  plotted  in  Fig. 


RELATIONS  OF  FLUX  E.M.F.  AND  CURRENT 


107 


59.  The  ordinates  are  the  volts  across  one  coil,  and  are  therefore  pro- 
portional to  the  flux  density.  Only  the  points  for  the  single,  two  and 
three-phase  connection  are  plotted.  The  points  for  the  six-phase  fall 
very  accurately  on  the  line  as  drawn,  and  are  omitted  to  avoid  con- 
fusion. 

It  will  be  seen  that  the  total  volt-amperes  are  practically  the  same  in 
all  of  the  cases.     It  is  true  that  the  volt-amperes  seem  a  little  more  in  the 


Thre  J 


P 

JafH 


•Total  Voli  Amperes 


1 


FIG.  59. — Volt  Amperes  Excitation  with  Single-,  Two-  and  Three-phase  Connections. 

case  of  the  two-phase  winding,  and  a  little  less  in  the  case  of  the  single 
phase.  The  difference  between  the  two  and  three  phase  is  possibly  on 
account  of  wave  shape,  although  the  wave  in  both  cases  approximated  a 
sine  shape.  The  connections  in  the  case  of  the  two-phase  and  the  single- 
phase  readings  was  the  same,  one  of  the  two  circuits  being  merely 
opened.  Operating  single-phase,  as  will  appear  later,  the  flux  ceases 
to  be  of  exactly  constant  value  in  the  different  positions,  being  somewhat 
weaker  when  at  right  angles  to  the  position  of  the  stator  winding. 
Hence  on  the  whole  a  slightly  lower  volt-ampere  excitation  will  be 
required. 


CHAPTER  VIII 


THE    LEAKAGE    COEFFICIENT    AND    THE    PREDETERMINATION 
OF  THE  CIRCLE  DIAGRAM 

IN  Fig.  60  is  given  a  simplified  form  of  the  circle  diagram.  It  is 
obtained  from  the  more  general  form  on  the  supposition  that  the  iron 
losses  are  negligible.  This  introduces  a  slight  error  in  our  calcula- 
tions but  allows  of  the  derivation  of  more  simple  formulae. 

The  predetermination  of  the  constants  of  the  circle  diagram  is  of 

the  greatest  importance  to  the 
designer.  In  its  essentials,  it 
consists  in  finding  the  lengths 
of  the  two  lines  OA  and  OB. 
The  former  of  these  is  the 
magnetizing  current  of  the 
motor,  or  it  is  the  current  the 
motor  would  take  if  it  were 
operating  at  synchronism  and 
if  it  had  no  losses.  Its  value 
is  slightly  less  than  that  of  the 

no-load  current.      The  determination  of  the  value  of  the  magnetizing 
current  has  been  treated  on  page  100. 

The  line  OB  represents  the  current  the  motor  would  take  if  it  were 
prevented  from  rotating  and  if  there  were  no  losses,  i.e.,  if  the  motor 
windings  had  no  resistance  and  the  iron  loss  were  zero.  In  the  case  of 
a  wound-rotor  machine,  if  we  neglect  the  losses,  the  current  OA  is  the 
current  taken  when  the  rotor  is  on  open  circuit  and  the  motor  at  rest. 
OB  is  the  primary  current  when  the  secondary  is  short  circuited  and 
is  prevented  from  rotating.  AB  is  the  rotor  current  under  the  same 
circumstances  on  the  supposition  that  we  have  an  equivalent  rotor, 
that  is  one  wound  with  the  same  number  of  turns  as  the  stator. 

The  determination  of  the  no-load  current  is  comparatively  simple, 
and  can  be  carried  out  with  a  considerable  degree  of  accuracy.  This 
is  due  to  the  fact  that  practically  all  of  the  flux  which  is  cut  by  the 

108 


O      A  D  B 

FIG.  60. — Simplified  Circle  Diagram. 


PREDETERMINATION  OF  THE  CIRCLE  DIAGRAM 


109 


stator  conductors  passes  directly  across  the  gap  and  is  cut  by  the 
rotor  conductors  as  well.  The  reluctance  of  the  part  of  the  path  which 
is  in  the  iron  is  comparatively  low,  and  may  frequently  be  neglected. 
The  path  of  the  flux  in  the  air  is  quite  definite  both  in  length  and  in 
area,  and  almost  the  only  complication  arises  from  the  fact  that  in 
addition  to  the  current  in  the  stator  conductors  we  have  also,  even  at 
no  load,  a  current  in  the  rotor  conductors. 

The  estimation  of  the  stator  current  with  locked  rotor  is  by  no  means 
so  simple.  This  is  due  to  the  fact  that  a  great  part  of  the  flux  in  this 
case  does  not  follow  the  path  previously  indicated.  In  Fig.  61  is 
shown  a  portion  of  a  stator  and  rotor.  The  arrows  drawn  in  the  slots 
represent  to  scale  the  value  of  the  currents  in  the  respective  conductors 
at  a  given  moment.  Since  the  rotor  is  prevented  from  turning  there 


FIG.  61. — Leakage  Fluxes  of  Induction  Motor. 

will  be  produced  in  it  a  large  current.  This  current  will  be  nearly 
equal  and  opposite  to  that  in  the  stator.  The  arrows  drawn  opposite 
the  rotor  slots  represent  to  scale  the  value  of  the  rotor  currents.  Under 
these  circumstances  a  portion  of  the  flux  will,  as  before,  take  the  paths 
indicated  by  the  three  lines  marked,  a\,  a2,  and  ay  This  flux  is  due 
to  the  fact  that  the  stator  current  is  somewhat  greater  than  the  rotor 
current  and  is  directly  opposite  to  it  in  phase.  The  one  will  nearly 
offset  the  other  and  the  resulting  flux  is  small  in  proportion  to  the  current. 
A  little  consideration  will  show  that  the  arrangement  of  the  con- 
ductors is  substantially  that  shown  in  Fig.  62.  The  crosses  indicate 
currents  directed  from  the  observer  and  the  dots  currents  toward  him. 
Since  the  current  in  the  upper  row  of  conductors  is  somewhat  greater 
than  that  in  the  lower  row,  there  will  be  a  resultant  m.m.f.  tending  to 
set  up  a  flux  in  the  paths  a\  and  a2.  There  will,  however,  be  a  much 
stronger  m.m.f.  setting  up  a  flux  around  the  paths  b\  and  b2.  Also, 


110  THE  INDUCTION  MOTOP 

since  there  are  spaces  between  the  different  conductors  of  the  solenoid, 
fluxes  will  circulate  as  indicated  by  Ci  and  c2. 

In  the  diagram  of  the  actual  motor  the  paths  of  the  fluxes  are 
indicated  by  similar  letters.  As  was  mentioned,  the  m.m.f.  acting 
to  produce  a  flux  in  the  horizontal  direction  is  much  greater  than  that 
acting  vertically,  but  at  the  same  time  the  reluctance  in  the  path  of  the 
former  flux  is  far  less  than  in  that  of  the  latter.  Hence  this  flux  will 
not  be  as  large  as  might  be  at  first  thought.  It  will  be  greater  the 
narrower  the  slots  are  compared  with  the  teeth,  the  narrower  the  open- 
ing of  the  slots,  and  the  farther  the  conductors  are  from  the  air  gap. 
Relatively  to  the  flux  directly  across  the  gap,  it  will  be  greater  the 
greater  is  the  air  gap,  and  the  less  is  the  pole  pitch. 

In  addition  to  the  fluxes  mentioned,  there  will  also  be  a  leakage 
flux  surrounding  the  end  connections  of  both  the  stator  and  the  rotor. 


FIG.  62. — Simplified  Diagram  of  Leakage  Fluxes. 

The  part  surrounding  the  rotor  conductors  will  obviously  be  small 
in  the  case  of  a  squirrel-cage  machine. 

Of  these  fluxes,  those  marked  by  the  letter  a  form  the  useful  flux 
of  the  motor.  The  others  are  leakage  fluxes.  Of  these  c  is  known 
as  tooth  leakage,  b  as  tooth-tip  or  zig-zag  leakage,  and  that  around 
the  end  connectors,  as  end  connectors  or  coil  end  leakage.  Also, 
since  the  currents  are  distributed  in  bands,  and  in  the  case  of  a  wound- 
rotor  machine  these  bands  are  continually  shifting  with  respect  to  one 
another,  there  will  be  set  up  a  pulsating  flux  surrounding  more  or  less 
of  the  conductors  of  these  bands.  This  is  known  as  belt  leakage. 
This  particular  leakage  thus  varies  with  the  relative  positions  of  the 
stator  and  rotor.  The  others  are  nearly  constant.  The  belt  leakage 
is  only  about  one-third  as  large  in  squirrel-cage  as  in  wound-motor 
machines. 

It  will  be  noted  that  the  total  flux  cutting  the  conductors  of  the 
stator  is  the  same  whether  the  motor  is  running  without  load,  or  is 
stationary  with  the  rotor  locked.  This  arises  from  the  fact  that  we 


PREDETERMINATION  OF  THE  CIRCLE  DIAGRAM  111 

are  considering  what  would  happen  if  the  circuits  were  without  resist- 
ance. Under  this  condition,  the  counter  e.m.f.  generated  in  the  stator 
must  at  all  times  equal  the  applied  e.m.f.  This  requires  with  a  con- 
stant e.m.f.  that  the  flux  be  constant. 

At  no  load,  this  constant  flux  consists  almost  entirely  of  the  flux 
marked  a,  and  since  it  is  confined  to  a  definite  path,  it  is  comparatively 
simple  to  calculate  the  flux  per  ampere,  or  conversely  the  current  needed 
for  a  given  flux.  With  locked  rotor  we  have  likewise  to  calculate  the 
flux  per  ampere,  since  having  this  value,  we  can  at  once  calculate  the 
current  required  for  the  total  flux,  or  the  length  of  the  line  OB,  Fig.  60. 
It  will  at  once  be  apparent  that  this  is  a  far  more  difficult  task  than  was 
the  calculation  of  the  no-load  current.  The  student  will  see  the  inherent 
possibility  of  laying  out  the  paths  of  the  various  fluxes,  and  computing 
the  value  of  each  per  ampere,  but  he  will  also  recognize  the  enormous 
difficulty  of  the  task  if  anything  like  a  general  solution  is  attempted. 
If  a  general  equation  could  be  derived  it  would  contain  among  others 
the  following  variables:  the  pole  pitch,  the  length  of  core,  air  gap, 
number  and  dimensions  of  slots,  permeability  of  the  iron  used,  type  and 
length  of  end  connections,  number  of  phases,  character  of  the  frame, 
etc.  It  is  evident  that  to  develop  or  use  such  a  formula  would  be  almost 
impossible. 

In  practice,  all  of  the  formulas  proposed  ignore  many  of  these 
factors.  It  is  tacitly  assumed  that  some  of  the  proportions  will  be 
about  the  average  of  other  motors  of  similar  size,  and  in  case  of  abnormal 
proportions  in  one  or  more  parts,  the  designer  is  expected  to  use  his 
judgment  to  modify  the  value  obtained.  One  of  the  simplest  pro- 
posals is  indicated  in  Fig.  63.  These  curves  are  constructed  using 
as  abscissas  the  values  of  the  pole  pitch  in  inches,  and  as  ordinates 
the  values  of  the  lines  of  flux  per  ampere-turn  per  inch  of  embedded 
length  of  the  conductors.  For  the  free  length  of  the  conductors  we  may  \ 
take  the  value  of  one  line  of  flux  per  inch  per  ampere-turn,  for  the 
case  of  phase-wound  machines,  or  the  value  0.75  for  squirrel-cage 
machines. 

To  apply  this  to  the  case  of  an  actual  motor,  let  us  take  a  machine 
whose  stator  has  an  internal  diameter  of  17  ins.,  and  a  net  length 
of  6  ins.  It  has  48  stator  and  no  rotor  slots,  and  is  wound  with  48 
coils  of  ii  turns  each  of  No.  6  wire.  The  machine  is  wound  for  three- 
phase  current  and  is  delta  connected.  It  is  a  four-pole  machine  and 
consequently  the  pole  pitch  is  13.4  in.  From  the  curve  it  will  be  seen 
that  we  shall  have  about  1.7  lines  per  ampere-turn  per  inch  of  core. 


112 


THE  INDUCTION  MOTOR 


This  value  corresponds  to  slots  three-quarters  closed.    The  calcula- 
tion is  as  follows: 

Length  per  turn 42.0  ins. 

Embedded  length 12.0     " 

Free  length 30.0     " 

Wires  per  slot 22 

Turns  per  phase  on  two  poles 11X48-^-3-7-2  =  88 


3.5 

I3'0 

!,5 

H 
§   2.0 

I  1.5 
1 
1.0 

0.5 

\ 

\ 

\ 

\ 

S 

\ 

\ 

\ 

\ 

X 

Ns 

X 

x 

^x 

\ 

x 

s 

^s, 

v^ 

>" 

^> 

. 

"\, 

^ 

^ 

\. 

—  ., 

^ 



1  —  —  . 

8             10             13             14             16             18 
Pole  Pitch  in  Inches 

FIG.  63. — Relation  of  Flux  per  Ampere  Turn  per  Inch  and  Pole  Pitch. 


NOTE. — It  is  necessary  to  consider  two  poles  as  a  unit,  since  all  the 
fiux  which  passes  through  the  windings  of  one  pole  of  a  pair  must  also 
pass  through  the  windings  of  the  other  pole  of  the  pair.  Consequently, 
fiux  generated  by  the  windings  of  one  pole  cuts  the  windings  of  the  other 
pole  of  the  pair  and  sets  up  an  e.m.f.  in  them.  Therefore  in  estimating 
the  inductance  of  the  machine  two  poles  must  be  regarded  as  a  unit. 
The  e.m.fs.  of  the  various  pairs  of  poles  are  of  course  added  together. 

Flux  per  ampere  per  pole  =  88  X  i .  7  X 1 2  +  88  Xo .  7  5  X  30  =  3800, 
Z,=  2X88X38oo^-io8=o.oo67  henry. 


PREDETERMINATION  OF  THE  CIRCLE  DIAGRAM  113 

The  inductance  of  the  rotor  will  be  approximately  the  same  as  that  of 
the  stator,  and  the  total  inductance  may  therefore  be  taken  as  being 
0.0134  henry. 

The  machine  is  wound  for  440  volts  and  the  three  windings  being 
connected  in  delta  each  one  is  subjected  to  the  full  pressure  of  the 
line.  Moreover,  on  account  of  the  delta  connection,  the  line  current 
will  be  equal  to  V^  times  the  current  taken  by  each  phase.  Then, 

Line  current  =  —  | — ^- — =362  amperes. 

0-0134X271X25     ' 

From  actual  test  the  locked  current  of  this  machine  was  376  amperes. 
The  agreement  is  remarkably  good,  and  is  far  better  than  could  in 
general  be  expected  from  this  method. 

DETERMINATION  OF  THE  LEAKAGE  COEFFICIENT  a 

Most  authors,  in  attacking  the  problem  or  predetermining  the 
circle  diagram,  prefer  to  determine  the  ratio  of  the  magnetizing  and 
locked-rotor  currents.  Then  by  dividing  the  no-load  current  by  the 
value  of  this  ratio,  we  can  determine  the  value  of  the  locked  current. 
This  ratio  is  usually  designated  by  the  letter  a,  or  a  =  OA  -±AB.  This 
procedure  has  the  advantage  that  the  value  of  a  is  a  constant  of  a  given 
frame  and  slotting,  and  does  not  change  to  any  great  extent  with  changes 
of  winding,  number  of  phases,  frequency,  flux  density,  etc.  It  does, 
however,  change  for  a  change  in  the  number  of  poles.  If  a  method 
similar  to  that  just  given  be  used,  the  value  of  this  ratio  is  at  once 
determined  from  the  ratio  of  the  magnetizing  current  and  the  locked 
current,  minus  the  magnetizing  current. 

Of  all  methods  proposed  for  the  determination  of  the  value  of  o, 
perhaps  that  developed  by  H.  M.  Hobart  is  the  most  general.  In 
brief,  the  method  consists  in  determining  a  series  of  curves,  one  for 
each  value  of  H,  the  average  of  the  rotor  and  stator  slots  per  pole,  and 
for  each  value  of  t,  the  pole  pitch.  In  all  58  such  curves  are  given. 
Values  of  H  from  6  to  27  are  included,  and  values  of  t  from  15  to  40. 
This  covers  well  the  range  of  ordinary  practice.  In  each  curve  the 
abscissas  denote  the  gross  length  of  the  stator  and  rotor  core.  The 
ordinates  are  the  values  of  a.  On  each  curve  sheet  are  drawn  three 
curves  corresponding  to  air  gaps  of  0.8,  1.5  and  2.5  mm.  These  curves 
are  stated  to  apply  to  motors  having  open  or  semi-open  slots,  and  having 
phase-wound  rotors.  In  the  case  of  squirrel-cage  motors,  the  value 


114  THE  INDUCTION  MOTOR 

of  o  is  estimated  as  being  0.8  of  the  value  for  the  corresponding  phase- 
wound  machine. 

It  will  be  seen  that  this  method  is  perfectly  general,  and  if  a  sufficient 
number  of  curves  were  determined,  could  be  made  to  cover  any  case. 
To  do  this  completely  would  require  curves  taking  into  account  all 
the  possible  variations  of  about  fourteen  variables.  In  the  curves 
mentioned,  four  of  these  are  included.  To  extend  the  method  to  a 
greater  number  would  require  almost  a  prohibitive  number  of  curves. 
The  method  is  of  course  frankly  empirical,  but  should  give  good  results, 
even  taking  into  account  only  the  four  variables.  Applying  it  to  the 
case  of  the  motor  previously  mentioned  we  find  that  a  has  the  value 
0.032.  From  the  actual  test  we  find  that  0=13. 6 -1-376  =  0. 0361. 
Moreover,  since  the  slot  opening  is  small  we  should  be  inclined  to 
increase  the  value  found  from  the  curve  to  somewhat  more  than  0.032. 
Hence  the  agreement  in  this  particular  case  is  good.  In  numerous 
other  instances  the  method  has  been  found  to  give  results  that  are  very 
consistent  with  the  test  results. 

An  extended  discussion  of  the  various  leakage  fluxes  has  been  given 
by  Prof essor  Comfort  A.  Adams,  in  Vol.  I,  Transactions  of  the  Interna- 
tional Electrical  Congress,  St.  Louis,  1904,  and  Vol.  XXIV,  Trans- 
actions of  the  American  Institute  of  Electrical  Engineers.  These  results 
have  been  collected,  and  the  formulae  somewhat  simplified  by  I.  E. 
Hansen,  in  the  Electrical  World,  Vol.  XLIX,  No.  13.  The  formulas, 
while  somewhat  tedious  in  practice,  are  stated  to  give  very  good  results. 

Dr.  Hans  Behn-Eschenburg  has  contributed  a  paper  on  the  same 
subject  to  the  proceedings  of  the  Institution  of  Electrical  Engineers. 
He  develops  the  following  formula  for  the  estimation  of  the  leakage 
coefficient. 


in  which  d  is  the  depth  of  air  gap;  h  the  average  number  of  slots  per 
pole  for  the  stator  and  rotor;  /  the  pole  pitch;  b  the  width  of  the  active 
iron  of  the  stator  and  rotor;  and  x  the  average  width  of  slot  openings. 
All  dimensions  are  in  centimeters. 

Applying  this  to  the  machine  previously  discussed  we  find  (7=0.043, 
which  is  in  fair  agreement  with  the  value  0.0361  found  by  test. 

Further  valuable  information  in  reference  to  the  leakage  coefficient 
is  embodied  in  a  number  of  papers  by  Rudolph  Helmund.  Some  of 


PREDETERMINATION  OF  THE  CIRCLE  DIAGRAM  115 

these  may  be  found  in  the  Transaction  of  the  American  Institute  of 
Electrical  Engineers  as  follows:  Zig-zag  Leakage  of  Induction  Motors, 
Vol.  XXVI,  p.  1505;  Graphical  Treatment  of  the  Rotating  Field, 
Vol.  XXVII,  p.  1375- 

BEHREND'S  METHOD 

The  following  formula  for  the  determination  of  the  leakage  coefficient 
was  first  proposed  by  B.  A.  Behrend  in  the  Electrical  World  and-  Engi- 

d 
neer,  November  24,  1900.     The  formula  is  simply  a=C—  ,in  which  d  is 

the  depth  of  the  air  gap,  and  /  is  the  pole  pitch.  C  is  a  constant  for  any 
given  set  of  dimensions,  but  varies  with  a  change  in  the  design  of  the 
motor. 

If  for  a  we  substitute  its  value  —,  in  which  i0  is  the  magnetizing 

*j 
current  and  it  is  the  current  with  locked  rotor,  we  may  write 


To  show  that  this  formula  holds,  assuming  C  to  be  a  true  constant, 
it  is  necessary  to  prove  that  IQ  is  proportional  to  the  length  of  the  air 
gap,  and  inversely  proportional  to  the  pole  pitch,  and  that  the  value 
of  the  locked  rotor  current  is  not  changed  by  a  variation  in  the  values 
of  d  and  /. 

The  first  of  these  propositions  is  almost  self-evident,  and  in  any 
event  was  fully  discussed  in  deriving  the  expression  for  the  value  of 
the  magnetizing  current.  That  the  magnetizing  current  is  inversely 
proportional  to  the  pole  pitch  will  be  apparent  if  we  consider  one  pole 
of  each  of  two  motors,  in  one  of  which  the  pole  pitch  is  double  that  of 
the  other.  If  we  assume  that  the  number  of  conductors  is  the  same  in 
the  two  cases,  it  will  be  evident  that  the  flux  density  in  the  air  gap  of 
the  motor  B  is  half  that  of  A.  This  follows  since  the  total  flux  must 
be  the  same  in  the  two  cases.  This  being  the  case,  the  magnetizing 
current  of  B  must  be  half  that  of  A. 

To  show  that  a  change  in  either  of  these  two  dimensions  does  not 
affect  the  value  of  the  locked  current  is  not  quite  so  simple.  Referring 
to  Fig.  61,  however,  it  will  be  seen  that  a  change  in  the  length  of  the 
air  gap  will  have,  at  most,  a  minor  effect,  since  the  path  of  any  of  the 


116  THE  INDUCTION  MOTOR 

fluxes  shown  there  will  be  little  affected.  It  is  true  it  has  some  influence, 
and  in  general  an  increase  in  the  length  of  the  gap  tends  to  increase  the 
locked  current  and  hence  to  increase  the  maximum  output  of  the  motor. 
Too  great  an  increase  in  the  air  gap  would  result  in  all  of  the  flux  being 
leakage  flux,  and  the  motor  would  of  course  have  no  output. 

That  the  locked  current  does  not  change  with  a  change  in  the 
pole  pitch  is  only  approximately  true.  The  statement  is  equivalent 
to  saying  that  the  leakage  flux  per  ampere  is  unchanged  for  a  change  in 
the  pole  pitch.  Of  the  three  fluxes  represented  in  Fig.  61,  b\  and  b2 
will  be  somewhat  decreased,  since  the  path  of  the  lines  is  somewhat 
increased,  unless  we  assume  that  the  dimensions  of  the  slots  are 
unchanged,  and  that  all  the  reluctance  is  in  the  gaps  between  the  teeth 
and  in  the  air  gap,  in  which  case  the  flux  would  be  unchanged.  The 
tooth  leakage  would  be  practically  the  same  with  the  same  slots,  or 
it  would  be  lessened  if  the  slots  were  increased  in  size  to  correspond 
with  the  increase  in  the  pole  pitch.  The  end  connector  leakage  would 
not  be  very  materially  affected,  since  the  lengthening  of  the  end  con- 
nectors would  tend  to  increase  it  and  at  the  same  time  the  lessened 
bunching  of  the  connectors  would  decrease  it.  If,  however,  too  great 
lengths  of  pole  pitch  are  used,  the  end  connector  leakage  becomes  of 
relatively  great  importance,  and  makes  the  design  poor. 

It  will  be  apparent  from  the  above  that  the  expression  as  given 
will  be  only  approximately  true,  if  we  regard  C  as  a  true  constant.  In 
order  that  the  formula  should  be  of  universal  application  it  will  be 
necessary  to  have  some  means  of  estimating  the  value  of  C.  This  value 
is  evidently  influenced  by  such  factors  as  the  shape  of  the  slots,  the 
type  of  winding,  the  length  of  the  core,  etc.,  but  as  was  pointed  out, 
even  the  factors  included  in  the  formula  have  an  influence  on  it. 

In  the  Electrical  World  and  Engineer  for  April  30,  1904,  Mr. 
H.  M.  Hobart  published  curves  for  determining  the  value  of  C.  These 
curves  take  into  account  the  ratio  of  pole  pitch  to  core  length,  and  the 
product  of  the  average  slots  per  pole  times  the  depth  of  the  air  gap. 
Curves  are  also  given  both  for.  open  and  for  completely  closed  slots. 
Values  for  partially  closed  slots  can  be  obtained  by  interpolation. 
The  curves  were  derived  empirically  from  the  results  of  tests  upon  a 
number  of  motors. 

In  "  Alternating  Current  Motors,"  by  A.  S.  McAllister,  it  is  shown 
that  these  curves  can  be  replaced  by  straight  lines,  and  that  values  taken 
from  the  straight  line  curves  agree  equally  as  well  with  the  facts  as 
do  those  taken  from  the  original  curves.  The  equations  of  these  lines 


PREDETERMINATION  OF  THE  CIRCLE  DIAGRAM  117 

are  readily  obtained,  and  it  is  found  that  the  value  of  C  can  be  repre- 
sented by  the  following  equation: 


A/  0.0972 

5.5^0+  2-9  rj  (0.54+  -u-, 


where  S0  is  the  percentage  opening  of  slots  (average  of  stator  and  rotor) 
/  the  pole  pitch  (or  in  the  case  of  short-pitch  windings,  coil  pitch), 
d  the  depth  of  air  gap  in  inches,  and  h  the  average  number  of  stator 
and  rotor  slots  per  phase  per  pole.  If  all  dimensions  are  in  centimeters, 
replace  0.0972  by  0.247.  In  tne  case  °f  short-pitch  windings  on  account 
of  the  overlapping  of  the  phases,  the  number  of  slots  containing  windings 
of  a  given  phase  is  greater  than  would  be  the  case  if  the  winding  were 
of  full  pitch.  It  is  approximately  correct  to  count  as  the  number  of 
slots  per  phase  per  pole,  the  actual  number  of  slots  per  pole  in  which 
any  of  the  windings  of  a  given  phase  occur.  The  fact  that  some  of  the 
slots  contain  windings  of  two  different  phases  need  not  be  considered. 
All  dimensions  in  the  above  equation  are  in  inches. 

Let  us  consider  for  a  moment  the  nature  of  the  constant  C.     We 
have  the  equation, 


and  we  have  previously  shown  that 


where  y  is  the  number  of  phases  and  for  exactness  should  be  a  large 
number.     Also, 

E 


where  L=LP+L8=  the  local  leakage  inductance  per  phase  of  the  primary 
and  secondary.     Also  we  have  shown  that 


m~     ANlPf 
Making  these  substitutions  we  get 


118  THE  INDUCTION  MOTOR 

If  now  we  designate  by  LI  the  inductance  of  the  motor  with  locked 
rotor  per  pole  per  phase,  per  turn,  per  cm.  length  of  the  stator,  we 
get 


or,  the  constant  C  is  the  inductance  corresponding  to  one  turn  of 
the  stator  divided  by  the  length  of  the  stator  iron  and  the  number  of 
phases  and  multiplied  by  7.86Xio8,  or  in  other  words  C  is  equal 
to  7.86  times  the  lines  of  induction  set  up  per  turn,  per  pole,  per  cm. 
length,  per  ampere  of  the  stator  with  the  rotor  locked,  divided  by  the 
number  of  phases. 

That  this  value  will  tend  to  be  approximately  a  constant  will  be 
apparent  at  once.  Further,  if  we  let  L%  be  the  inductance  per  turn 
per  pole  of  the  stator  and  consider  for  the  moment  that  the  factor 

has  the  value  one,  we  can  write 


dh 


From  this  it  appears  that  the  inductance  is  proportional  to  a  constant 
times  the  length  of  the  stator  core,  the  value  of  this  constant  depending 
upon  the  type  of  slot  used,  plus  a  constant  times  the  length  of  the  pole 
pitch  or  a  different  constant  times  the  length  of  the  end  connector,  and 
times  the  number  of  phases.  We  may  consider  the  other  portion  of 


. 

the  expression  (0.54  +  —     —  )  as  a  factor  modifying  the  value  of  the 
\  ok    I 

above-mentioned  constants  to  a  certain  extent  depending  upon  the 
length  of  the  air  gap  and  the  number  of  slots  per  phase  per  pole.  We 
thus  see  that  there  is  a  definite  physical  basis  for  the  value  of  C  as 
derived  from  the  equation.  In  fact  the  method  in  its  ultimate  analysis 
reduces  to  almost  the  same  thing  as  the  method  first  explained.  The 
principal  point  of  difference  is  in  the  modification  of  the  constants  for 
the  lines  per  inch  per  turn,  depending  upon  the  opening  of  the  slots, 
the  air  gap,  and  the  number  of  slots  per  phase  per  pole.  The  appli- 
cation of  the  method  is  of  course  much  more  simple  than  that  of  the  first 
method. 

Let  us  apply  this  to  the  case  of  the  machine  before  mentioned. 
As  before  stated,  S=  -J,  ^  =  34  cm.    In  this  case,  however,  the  coil  pitch 


PREDETERMINATION  OF  THE  CIRCLE  DIAGRAM  119 

is  from  one  to  ten,  whereas  full  pitch  would  be  from  one  to  thirteen. 
For  the  value  of  t  in  the  formula  we  therefore  use  34X1^  =  25.5.  The 
number  of  slots  per  phase  per  pole  in  the  stator  is  48-4-3-7-4=4.  On 
account  of  the  short  pitch,  however,  the  coils  of  each  phase  are  dis- 
placed side  wise  three  slots,  and  we  therefore  take  as  the  number  of 
slots  per  phase  per  pole  in  the  stator  the  value  4  plus  3  or  7.  In  the 
rotor  we  have  110-4-12  =  9.18,  giving  as  the  average  of  both  the  stator 
and  the  rotor  8.09.  Substituting  these  values  in  the  formula  we  find 
C  to  be  8.95.  This  value  of  C  would  indicate  a  starting  current  of 
386  amperes.  The  actual  starting  current,  as  before  noted,  was  376. 
It  is  therefore  in  very  fair  agreement. 

Using  any  of  the  methods  just  explained,  we  can  determine  all  the 
values  necessary  to  enable  us  to  lay  out  the  circle  diagram,  and  from  this 
diagram  we  can  obtain  the  value  of  the  maximum  input  to  the  motor. 
It  is  highly  desirable,  however,  to  have  a  means  of  determining  directly 
the  value  of  the  maximum  input,  and  consequently  approximately  the 
maximum  output  of  the  motor.  This  we  can  readily  do  by  making 
use  of  the  expression  for  the  value  of  C  which  we  have  just  derived. 

If  E  equals  the  e.m.f.  per  phase,  and  there  are  y  phases,  we  have, 
letting  L  equal  the  inductance  per  phase, 

E      Ey          E2y 

maximum  k.w.  input  =P =— —  •  — —  =  — — =-— •. 
4K/L  1000     40007I/L 

Also  we  have  shown  that  for  large  values  of  y  we  have 

7.86Xio8£  =  7.86Xio8/>£ 
ypbNi2    ~~         ybN2       ' 

Substituting  this  in  the  expression  for  the  maximum  input,  we  have 


CbfN2    ' 
or,  in  inch  units, 

246oo/>£2 
CbfN2  ' 

As  noted,  the  above  expression  for  C  was  developed  on  the 
assumption  that  the  number  of  phases,  y,  was  large.  In  other  words, 
no  account  was  taken  of  the  fact  that  the  e.m.f.  generated  in  the  various 
conductors  in  the  winding  of  a  phase  are  not  in  the  same  phase.  To 
allow  for  this,  we  must  multiply  the  sum  of  all  the  e.m.fs.  generated  in 


120  THE  INDUCTION  MOTOR 

the  windings  of  a  phase  by  the  so-called  breadth  coefficient.  This  was 
fully  explained  on  page  97.  In  this  case,  then,  if  we  have  a  three- 
phase  winding  we  must  divide  the  above  constant  by  0.953,  and  for  a 
two-phase  winding,  by  0.90.  This  gives  us  the  following  values: 


P  (three  phase)  = 


P  (two  phase)  = 


In  the  case  of  short-pitch  windings,  another  correction  must  be 
made.  As  was  shown,  the  use  of  the  fractional-pitch  winding  has  the 
same  effect  as  reducing  the  number  of  conductors.  To  make  this 

correction,  we  multiply  the  number  of  conductors  by  sin  —  .     In  the 

formula  for  the  maximum  output,  the  same  factor  should  be  introduced, 
which  gives  us  as  the  expression  in  the  case  of  fractional  pitch 


P  (three  phase)  =  - 


Considering  the  above  formula  it  will  be  seen  that  the  output  is 
proportional  to  the  square  of  the  applied  e.m.f.,  and  inversely  pro- 
portional to  the  square  of  the  number  of  turns  (or  conductors)  in  the 
complete  winding.  These  relations  are  obvious  in  the  light  of  what 
has  already  been  stated.  Nothing  else  being  changed,  the  input  is  pro- 
portional to  the  number  of  poles,  and  inversely  proportional  to  the 
frequency  and  to  the  width  of  the  motor.  The  truth  of  the  former  will 
be  at  once  apparent  if  we  consider  the  motor  as  a  transformer.  The 
maximum  current  that  could  be  forced  through  the  motor  assuming  no 
resistance  in  the  circuits  would  be  inversely  as  the  frequency,  the  e.m.f., 
number  of  turns,  etc.,  being  kept  constant.  Similar  reasoning  will 
show  that  as  the  width  of  the  motor  increases,  the  inductance  will  in- 
erease  and  consequently  the  maximum  current  will  decrease. 

The  above  form  of  this  formula  is  perhaps  the  most  useful,  but 
by  various  substitutions  we  can  change  it  into  a  variety  of  expressions 
which  may  perhaps  help  to  make  clear  some  of  the  actions  of  the 


PREDETERMINATION  OF  THE  CIRCLE  DIAGRAM  121 

induction  motor.     Thus  in  the  above,  if  we  substitute  for  E  its  value 
when  the  number  of  phases  is  very  great, 


p     1.41 
'    E=~ 


we  obtain, 


If  we  let  Ac,  the  total  cylindrical  surface  of  the  inside  of  the  stator 
equal  pA,  and  Sp,  the  peripheral  speed  of  the  rotating  flux,  equal  10 
//,  we  obtain: 

4.92-VW 
io13C 

or,  the  maximum  input  of  the  motor  is  equal  to  a  constant  times  the 
peripheral  speed  of  the  motor  at  synchronism,  times  the  area  of  the 
inside  of  the  stator,  times  the  square  of  the  maximum  value  of  the  flux 
density  in  the  air  gap,  divided  by  the  value  of  the  dispersion  coefficient. 
These  relations  are  almost  self  evident. 

If  we  neglect  the  loss  in  the  stator  conductors,  and  assume  that 
all  of  the  input  appears  as  torque  at  the  rotor  surface,  we  may  equate 
the  maximum  input  with  the  work  done  on  the  rotor,  and  letting  p 
denote  the  pull  per  square  inch  of  the  stator  surface,  we  may  write  , 


^ 

io13C  33°°° 

This  readily  reduces  to 

,     2.18  B 


The  interpretation  of  this  equation  is  that  for  each  value  of  the  con- 
stant C,  there  exists  a  definite  relation  between  the  maximum  pull 
per  square  inch  and  the  value  of  the  flux  density  in  the  air  gap.  This 
relation  is  of  course  evidently  true. 

This  same  relation  enables  us  to  obtain  a  rather  curious  interpre- 
tation of  the  constant  C.  The  value  of  p  which  we  have  just  obtained 
is  the  maximum  attainable  value  of  the  tangential  pull  per  square  inch 
of  the  stator  surface.  The  value  of  the  radial  pull  is  obtained  from 
the  equation: 


122  THE  INDUCTION  MOTOR 

in  which  pr  is  the  radial  pull  due  to  the  flux  density  Bm  per  square  inch. 
Before  comparing  these  we  must  remember  that  p  is  the  average  pull 
all  over  the  surface  of  the  stator.  The  maximum  pull  will  be  greater 
in  the  ratio  of  the  square  of  the  maximum  value  divided  by  the  average 

value.  This  ratio,  as  we  have  shown,  is  equal  to  —  .  If  then  pm  is  the 
maximum  value  of  the  tangential  pull  per  square  inch,  we  have, 


Letting  r  be  the  ratio  of  these  two  quantities,  we  have 

T=A=^_- 
Pm     3-88' 
or, 

C     i  8«  pr 
'=3'88K 

It  thus  appears  that  C  may  be  interpreted  as  3.88  times  the  radial 
pull  per  unit  area,  divided  by  the  maximum  tangential  pull  per  unit 
of  area. 


CHAPTER  IX 

SOME  GENERAL  CONSIDERATIONS  RELATING  TO  DESIGN 

HIGH-   VS.   LOW-FREQUENCY   MOTORS 

A  NUMBER  of  the  facts  previously  discussed  have  reference  prin- 
cipally to  the  design  of  the  induction  motor.  It  is  the  intention  to 
group  in  this  chapter  a  number  of  facts  and  formulas  of  interest  prin- 
cipally to  the  designing  engineer,  rather  than  to  the  operating  engineer. 

The  question  of  whether  a  high-  or  a  low-frequency  current  is  pref- 
erable for  operating  induction  motors  is  one  which  can  be  answered 
in  different  ways,  depending  upon  the  circumstances.  In  this  country 
the  choice  usually  lies  between  60  and  25  cycles.  In  the  following 
discussion,  however,  the  frequencies  of  50  and  25  cycles  will  be  con- 
sidered on  account  of  the  greater  ease  of  making  comparisons.  In 
general  we  might  make  the  statement  that  for  small  motors,  taking 
everything  into  consideration,  the  higher  frequency  is  preferable,  and 
on  the  other  hand,  for  large  motors,  the  lower  frequency  is  the  more 
desirable.  Another  way  of  stating  the  same  thing  is  to  say  that  for 
the  same  speed,  oj  rotation,  the  low-frequency  motor  will  have  the  better 
electrical  characteristics.  It  will,  however,  usually  be  somewhat 
higher  in  price.  We  can  perhaps  best  establish  the  truth  of  these 
statements  by  examining  a  couple  of  typical  cases. 

Consider  first  the  case  of  a  io-h.p.,  three-phase,  5o-cycle,  44o-volt 
motor.  A  machine  for  this  frequency,  (in  all  sizes  from  io-h.p.  down) 
would  usually  be  wound  for  four  poles,  giving  a  synchronous  speed 
of  1 500  rev.  per  min.  This  is  assuming  that  the  motor  is  for  ordinary 
belted  service.  For  special  applications,  many  other  speeds  may  be 
needed.  For  2 5 -cycle  service  the  speed  adopted  for  all  sizes  from  50 
h.p.  down  would  in  most  cases  be  750  rev.  per  min.  Of  course  these 
motors  could  be  wound  with  two  poles,  giving  a  speed  of  1500  rev.  per 
min. ,  but  this  is  rarely  done.  The  objection  to  doing  so  is  that  since 
the  flux  per  pole  is  twice  as  large  as  in  the  four-pole  type,  the  section 

123 


124  THE  INDUCTION  MOTOR 

of  iron  back  of  the  slots  must  be  twice  as  great,  for  the  same  rotor 
diameter.  Moreover,  the  end  connections  are  very  long  and  the  machine 
is  difficult  to  wind.  On  account  of  these  facts,  the  two-pole  machine 
is  rather  rarely  used  except  in  the  very  small  sizes.  This  inherently 
lower  speed  is  the  fundamental  reason  why  the  low-frequency  machine 
is  less  satisfactory  in  the  small  sizes  than  the  high-frequency  one. 

To  study  this  more  in  detail,  let  us  consider  the  case  of  the  io-h.p., 
5o-cycle,  i5oo-rev.  per  min.,  44o-volt  machine  mentioned  above.  If 
it  were  desired  this  same  machine  could  be  operated  without  change 
upon  a  25-cycle,  2  20- volt  circuit.  The  motor  might,  it  is  true,  be 
slightly  improved  by  somewhat  increasing  the  flux  density,  but  for 
our  purposes  we  may  assume  that  no  change  is  made  in  it.  The  flux 
density  is  obviously  the  same  as  before,  since  the  frequency  and  the 
voltage  are  both  halved.  Hence,  the  torque  will  be  the  same  for  the 
same  current,  and  since  the  speed  is  only  half  as  great,  the  rating  will 
now  be  5  h.p.  The  first  great  disadvantage  of  the  low-frequency 
machine  is  at  once  apparent.  The  cost  of  a  5-h.p.,  25-cycle  machine 
is  practically  the  same  as  the  cost  of  a  10  h.p.,  5o-cycle  machine.  It 
is  true  that  if  a  resistance-type  starter  is  used,  something  may  be  saved 
in  the  cost  of  this  item,  but  if  an  auto-starter  is  used,  the  cost  of  the 
5-h.p.,  25-cycle  starter  will  be  practically  the  same  as  the  cost  of  one 
for  the  io-h.p.,  5o-cycle  motor.  The  fact  that  the  machine  operates 
at  a  lower  speed  is  in  itself  a  slight  advantage,  and  will  lead  to  more 
quiet  operation  and  lower  maintenance  charges.  However,  giving 
these  facts  their  full  value,  it  is  still  true  that  to  equip  a  factory  with 
25-cycle  motors  of  small  and  medium  sizes,  costs  far  more  than  would 
be  the  case  if  they  were  to  operate  on  50  cycles. 

To  deduce  the  comparative  electrical  properties,  consider  the  circle 
diagrams  of  the  two  machines.  Since  the  frequency  is  half  as  great 
in  the  one  case  as  in  the  other,  and  since  the  proper  voltage  is  also 
approximately  half  as  great,  the  diameter  of  the  circle  will  be  the  same 
in  the  two  cases.  Hence  it  will  be  at  once  apparent  that  the  power- 
factor,  the  overload  capacity  and  the  starting  torque  will  be  the  same. 
The  overload  capacity  and  the  starting  torque  are  of  course  supposed 
to  be  expressed  in  percentage  of  the  full-load  torque. 

To  compare  the  efficiencies,  let  us  assume  that  in  the  5o-cycle  motor 
the  iron  loss  is  500  watts,  the  copper  loss  500,  and  the  friction  loss  200 
watts.  In  the  25-cycle  machine  the  iron  loss,  since  the  density  is  the 
same,  will  be  somewhat  less  than  half,  say  225  watts,  the  copper  loss 
will  be  unchanged,  and  the  friction  loss  will  also  be  less  than  half,  say 


HIGH-  VS.  LOW-FREQUENCY  MOTORS  125 

75  watts.     The  efficiency  of  the  5o-cycle  motor  is  then  86.2  per  cent 
that  of  the  other,  82.2  per  cent. 

It  would  be  fairer  to  compare  motors  of  the  same  horse-power 
rating,  and  if  this  were  done  the  discrepancy  in  efficiency  would  not 
be  so  great.  The  power-factor  of  the  low-frequency  motor  would  prob- 
ably be  somewhat  higher.  The  cost,  while  much  greater,  would  by  no 
means  be  twice  as  great.  The  actual  ratio  at  present  prices  is  about 
in  the  proportion  of  i  to  \/2  or  1.41. 

When  we  consider  motors  large  enough  so  that  the  speed  is  limited 
by  other  factors  than  the  characteristics  of  the  motor,  the  result  is  quite 
different.  Consider  the  case  of  a  loo-h.p.  motor.  Such  a  machine 
would  probably  be  wound  to  operate  at  5oo-rev.  per  min.  for  either 
frequency.  In  this  case  all  the  advantage  except  in  the  point  of  cost 
is  with  the  low-frequency  motor.  That  the  cost  of  the  25-cycle  machine 
is  higher  is  due  to  the  fact  that  since  each  pole  is  twice  the  size  of  those 
of  the  other  motor,  the  section  of  iron  to  carry  the  flux  must  be  greater. 
It  is  usually  not  twice  as  great,  since  the  density  may  be  greater  on 
account  of  the  lower  frequency,  but  it  is  materially  more.  The  end 
connections  are  likewise  about  twice  as  long,  and  consequently  the 
cost  of  copper  is  greater.  For  much  the  same  reasons  the  cost  of  the 
auto-starter  is  also  greater.  The  ratio  of  costs  at  the  present  time  is 
about  as  i  to  i.i. 

In  comparing  the  electrical  characteristics  of  the  two  machines, 
we  may,  for  the  sake  of  simplicity,  assume  them  to  have  rotors  of  the 
same  diameter  and  length.  The  tendency  in  practice  would  be  to  make 
the  25-cycle  machine  of  a  smaller  diameter  and  a  greater  length  than 
the  5o-cycle  motor.  The  comparison  is  therefore  somewhat  unfair 
to  the  machine  of  lower  frequency.  We  may  likewise  assume  for  the 
sake  of  simplicity  that  the  flux  per  square  inch  in  the  air  gap  is  the 
same  in  both  cases,  and  that  the  air  gap  is  the  same. 

The  first  noticeable  point  is  that  since  the  ampere-turns  needed  to 
maintain  the  flux  across  the  air  gap  depend  only  upon  the  flux  density 
and  the  length  of  the  gap,  the  ampere-turns  per  pole  will  be  the  same  in 
the  two  cases.  Since  the  low-frequency  motor  has  only  half  as  many 
poles  as  the  other,  the  total  ampere-turns  will  be  only  half  as  great. 
Hence  the  no-load  current  will  be  only  half  of  that  taken  by  the  5o-cycle 
motor.  As  can  readily  be  seen  from  the  circle  diagram,  the  power- 
factor  of  the  low-frequency  motor  will  be  much  higher.  The  fact  that 
the  leakage  path  from  pole  to  pole  is  twice  as  great,  improves  the  leakage 
coefficient  and  likewise  the  power-factor  of  the  low-frequency  machine. 


126  THE  INDUCTION  MOTOR 

Since  its  circle  diagram  is  so  much  larger,  the  pull-out  point,  the  starting 
torque,  and  the  point  of  maximum  power-factor  will  all  be  greater. 

Whether  or  not  the  efficiency  will  be  greater  depends  upon  a  number 
of  features.  The  fact  that  the  frequency  is  lower  will  tend  to  make 
the  iron  loss  less,  but  on  the  other  hand  the  copper  loss  will  be  greater 
on  account  of  the  longer  end  connections.  Probably  the  efficiency 
will  not  be  greatly  different  in  the  two  cases.  In  the  above  it  must  be 
remembered  that  full  credit  has  not  been  given  to  the  25-cycle  machine, 
since  we  have  assumed  that  it  followed  the  design  of  the  5o-cycle  motor. 
It  could  be  improved  in  a  number  of  particulars  by  various  modifica- 
tions in  the  design.  However,  in  any  case  it  is  obviously  the  better 
machine  of  the  two  from  the  electrical  standpoint. 

OPEN  OR  CLOSED  SLOTS 

In  this  country  the  completely  closed  slot  is  rarely  used.  This 
is  on  account  of  the  excessive  cost  of  winding  such  a  motor.  The  wires 
must  be  threaded  through  one  at  a  time  from  the  end,  and  the  process 
is  necessarily  tedious  and  expensive.  Abroad,  where  labor  cost  is 
less,  such  slots  are  sometimes  used;  in  this  country  as  far  as  the  writer 
knows,  they  are  never  employed.  When,  therefore,  closed  slots  are 
referred  to,  partially  closed  slots  are  meant.  In  these  slots  an  opening 
is  left  about  twice  as  wide  as  the  thickest  wire  to  be  used.  The  coils 
are  first  wound  up  on  a  form,  and  a  short  distance  at  each  end  is  then 
taped,  so  as  to  hold  the  coil  in  shape.  The  slot  insulation,  consisting 
of  paper,  oiled  muslin,  etc.,  is  placed  in  position,  and  the  wires  of  the 
coil  are  slipped  into  the  slot,  one  at  a  time.  The  projecting  parts  are 
taped  after  the  coil  is  in  place. 

In  the  case  of  the  wide,  open  slots,  on  the  other  hand,  the  coils  are 
completely  formed,  taped,  in  some  cases  impregnated  with  insulating 
material,  and  are  then  placed  in  position.  The  slot  lining  in  this  case 
is  very  thin,  being  merely  sufficient  to  protect  the  coil  from  injury  while 
it  is  being  placed  in  the  slot. 

Of  the  two  constructions,  there  is  little  doubt  that  from  the  cus- 
tomer's standpoint,  given  two  motors  of  identical  characteristics,  the  one 
with  open  and  the  other  with  closed  slots,  the  former  would  be  pre- 
ferred. This  is  on  account  of  the  fact  that  with  the  open  slots  a  better 
opportunity  is  presented  to  insulate  the  coil,  and  since  in  case  a  coil 
does  break  down,  it  can  be  replaced  somewhat  more  easily  than  would 
be  the  case  with  closed  slots.  The  difference  in  the  time  of  replacement 


HIGH-  VS.  LOW-FREQUENCY  MOTORS 


127 


is  not ,  however,  as  great  as  might  be  thought,  since,  in  either  case,  a 
large  amount  of  time  is  spent  in  disconnecting  and  reconnecting  the 
coils,  and  this  time  is  the  same  with  either  open  or  closed  slots. 

In  many  cases  it  is  impossible  to  build  a  motor  with  open  slots 
with  as  good  characteristics  as  one  with  partially  closed  slots.  Thus, 
in  Fig.  64,  let  the  full-line  curve  represent  the  circle  diagram  of  a  motor 
with  closed  slots.  If,  keeping  exerything  else  the  same,  the  slots  are 
made  open  instead  of  closed,  there  will  be  two  changes  apparent  in 
the  diagram.  In  the  first  place,  the  line  OA,  representing  the  mag- 
netizing current,  will  be  lengthened  to,  say,  OA'.  This  is  so  since  the 
flux  now  crosses  from  stator  to  rotor  in  tufts  instead  of  being  nearly 
uniform.  The  flux  density  in  the  tufts  is  of  course  greater  than  would 
be  the  case  if  the  iron  were  unbroken,  and  the  magnetizing  current  will 


\ 


FIG.  64. — Effect  of  Open  Slots  upon  Circle  Diagram. 

be  correspondingly  increased.  The  other  effect  is  that  the  diameter 
of  the  circle  will  be  greater.  This  is  so,  since  on  account  of  the  open 
slots  the  tooth-tip  leakage  is  less,  and  consequently  the  impedance  of 
the  motor  is  less.  The  corresponding  circle  diagram  of  the  open-slot 
motor  is  shown  by  the  dotted  lines. 

It  is  evident  that  in  the  open-slot  motor  the  power-factor  is  lower 
at  light  loads  and  higher  at  heavy  loads.  The  maximum  power- 
factor  may  be  better  or  worse.  If  the  diameter  of  the  circle  is  increased 
in  a  greater  proportion  than  the  magnetizing  current,  it  will  be  better, 
and  vice  versa.  If  the  two  are  increased  in  the  same  proportion, 
there  will  be  no  change  in  the  maximum  power-factor.  In  most  cases' 
if  the  closed-slot  motor  were  designed  so  that  its  characteristics  were 
the  best  that  could  be  secured  with  the  given  core,  those  of  the  open- 
slot  motor  would  be  poorer.  Thus  the  maximum  value  of  the  power- 
factor  will  come  at  too  large  a  load  and  will  be  lower  throughout  the 
operating  range  of  the  motor.  On  account  of  the  greater  current 


128  THE  INDUCTION  MOTOR 

required,  this  will  result  in  a  lowering  of  the  efficiency  of  the  motor 
throughout  its  normal  range,  and  a  slight  improvement  at  overloads. 
It  is  true  that  the  pull-out  point  and  the  starting  torque  will  be  higher, 
but  in  general  the  motor  will  be  slightly  inferior  to  the  closed-slot 
motor,  so  far  as  its  electrical  characteristics  are  concerned,  and  slightly 
superior  from  a  mechanical  standpoint. 

The  exception  to  this  is  the  case  of  motors  of  long  pole  pitch.  Such 
motors  are  usually  but  not  necessarily  low-frequency  motors.  The 
ampere-turns  required  to  maintain  the  flux  across  the  gap  are  propor- 
tional to  the  number  of  poles,  to  the  length  of  the  air  gap,  and  to  the  flux 
density  in  the  gap,  but  are  entirely  independent  of  the  length  or  width 
of  the  poles.  The  same  thing  is  true  of  any  electrical  machine,  either 
direct  or  alternating.  The  large  poles,  other  things  being  equal,  require 
more  copper  on  account  of  their  greater  size,  but  not  more  ampere-turns. 
Thus  compare  two  motors  wound  on  the  same  frame  for  the  same  speed 
and  the  same  output,  one  having  eight  poles  for  5o-cycle  current,  and 
the  other  four  poles  for  25-cycle  current.  The  two  motors  will  have 
approximately  the  same  full-load  current,  but  the  magnetizing  current 
of  the  25-cycle  machine  will  be  only  half  that  of  the  other. 

It  will  be  evident  that  in  the  case  of  such  machines  as,  for  example, 
large  four-pole,  75o-rev.  per  min.,  25-cycle  machines,  the  magnetizing 
current  will  be  a  comparatively  small  proportion  of  the  full-load  cur- 
rent. Hence  it  may  readily  happen  that  opening  the  slots  will  so  much 
increase  the  diameter  of  the  circle  as  to  offset  the  larger  magnetizing 
current.  This  is  especially  so  since  such  motors  usually  have  generous 
air  gaps,  and  opening  the  slots  has  then  a  comparatively  small  effect 
upon  the  magnetizing  current. 

The  above  considerations  will  also  serve  to  emphasize  the  difficulty 
of  designing  motors  for  low  speeds,  small  outputs  and  high  frequency, 
i.e.,  with  narrow  poles.  If  such  conditions  must  be  met,  the  only 
remedy  is  to  use  nearly  closed  slots,  and  the  shortest  possible  air  gaps. 
Such  machines  should  be  avoided  if  possible,  by  employing  back-geared 
motors,  belts  or  other  devices. 

BEST  DIAMETER  or  ROTOR 

It  is  obvious  that  in  designing  an  induction  motor  it  may  be  made 
of  large  diameter  and  short  along  the  shaft,  or  of  great  length  and  small 
diameter.  The  selection  of  the  most  appropriate  dimensions  is  of 
great  importance.  Like  most  other  problems  in  designing,  this  does 
not  admit  of  an  exact  solution,  since  so  many  antagonistic  factors  are 


HIGH-  VS.  LOW-FREQUENCY  MOTORS 


129 


involved.     We  can,  however,  point  out  the  considerations  which  should 
govern  the  designer  in  deciding  these  questions. 

The  first  point  to  be  kept  in  mind  is  that  the  product  of  the  square 
of  the  diameter  of  the  rotor  times  the  axial  length  of  the  stator  iron  is 
approximately  a  constant;  or 


This  is  based  upon  the  supposition  that  the  flux  density  and  the 
number  of  ampere-turns  per  inch  of  the  periphery  are  independent 
of  the  diameter  of  the  rotor.  While  it  might  be  advantageous  to 
change  these  quantities  slightly  as  the  diameter  of  the  rotor  was  changed, 
yet  the  assumption  is  nearly  exact.  Assuming  this  to  be  the  case,  it 
will  be  seen  at  once  that  the  output  of  the  motor  will  be  directly  pro- 
portional to  the  length  of  the  motor.  It  will  also  be  proportional  to 
the  square  of  the  rotor  diameter,  since  if  this  latter  is  doubled,  the  pull 
per  inch  of  periphery  will  be  unchanged,  but  the  periphery  will  be 
doubled,  and  the  -speed  of  the  moving  conductors  will  also  be  doubled. 
Consequently  the  output  of  the  machine  will  be  quadrupled,  or  it  will 
be  proportional  to  the  square  of  the  rotor  diameter. 


ECONOMY  OF  IRON 

The  amount  of  iron  required  for  the  core  is  not  affected  by  the 
diameter  of  the  rotor.  Thus  in 
Fig.  65,  let  i  represent  the  core 
of  an  induction  motor,  and  let 
2  represent  the  core  of  a  machine 
of  the  same  rating  but  of  a  dia- 
meter k  times  as  great.  It  will 
be  necessary  to  make  the  radial 
depth  of  iron  k  times  as  great, 
since  each  pole  is  k  times  as 
long,  and  the  extra  depth  is  " 
necessary  in  order  to  carry  the 
greater  flux  per  inch  along  the 
shaft.  This  necessitates  making  U-^ 

the    outside    diameter    and  the 
diameter  of  the  opening  in  the        FIG.  6S.-Relation  Betwee?  Weight  of 

.  Iron  and  Diameter  of  Core 

rotor  k  times  as  great.  The  length 

of  the  iron  along  the  shaft,    on  the  contrary,  will  need  to  be  only  — 


130  THE  INDUCTION  MOTOR 

times  as  much.  These  deductions  neglect  the  fact  that  the  depth 
required  for  the  teeth  does  not  greatly  change  with  the  diameter.  It 
will  be  readily  seen  that  the  volume  of  iron  in  2  is  equal  to 


and  it  will  be  apparent  at  once  that  the  same  expression  is  the  volume 
of  iron  in  i. 

If  we  were  to  take  account  of  the  fact  that  the  depth  required  for 
the  teeth  does  not  change,  we  should  find  that  there  is  a  slight  economy 
of  iron  by  using  the  greater  diameter. 


COPPER  ECONOMY 

As  far  as  the  active  copper,  i.e.,  the  copper  in  the  slots,  is  concerned, 
the  greater  the  diameter  the  less  the  amount  of  copper  required.  This 
is  so  since  with  a  greater  rotor  diameter,  each  element  of  the  winding 
is  cutting  the  same  flux  as  a  greater  velocity.  The  current,  carrying 
capacity  will  be  at  least  as  great,  and  consequently  the  output  per 
pound  of  copper  will  be  greater.  If,  however,  we  increase  the  diameter, 
keeping  the  number  of  poles  and  the  speed  constant,  the  pole  pitch,  and 
consequently  the  length  of  the  end  connections,  is  increased.  After  a 
certain  diameter  is  reached,  the  increase  in  the  copper  in  the  end  con- 
nections will  more  than  offset  the  saving  in  the  active  copper. 

It  is  not  difficult  to  compute  the  diameter  which  will  give  the  greatest 
economy  of  copper. 

.  Let  K  denote  the  weight  of  copper  per  inch  of  periphery  and  per 
inch  of  length  along  the  shaft;  p  the  pole  pitch  in  inches;  N  the  number 
of  poles;  b  the  length  of  the  iron  of  armature  parallel  to  the  shaft;  d 
the  mean  diameter  of  winding;  r  the  ratio  of  length  of  end  connectors 
to  pole  pitch,  and  W  the  total  weight  of  copper. 

Then, 


HIGH-  VS.  LOW-FREQUENCY  MOTORS  131 


Then  for  the  minimum  value  of  W, 
dW 


or, 

But,  since  P  —  J^,  substituting  for  N  and  k\  we  have 
b  =  2rp. 

Since  b  is  the  active  length  of  one  conductor  and  rp  is  the  inactive  length, 
for  minimum  copper  the  shape  of  the  armature  should  be  such  that 
the  active  wire  is  twice  the  inactive.  If  we  assume  that  the  end  con- 
nector is  1.5  times  the  coil  pitch  the  rule  may  be  stated  thus.  The  shape 
of  the  armature  for  the  minimum  amount  of  copper  should  be  such 
that  the  coil  pitch  is  one-third  of  the  active  length  of  the  armature  or 


The  above  applies  to  direct  as  well  as  to  alternating-current  genera- 
tors and  motors. 

DIAMETER  FOR  BEST  POWER-FACTOR 

In  the  above  we  have  shown  the  diameter  of  rotor  which  will  require 
the  smallest  amount  of  copper.  This  diameter  will  also  be  the  one  which 
will  give  nearly  the  highest  efficiency  or  the  lowest  losses.  This  will 
be  apparent  when  we  consider  that  the  amount  of  iron,  and  conse- 
quently the  iron  loss  is  independent  of  the  rotor  diameter,  and  since 
the  diameter  deduced  above  requires  the  smallest  amount  of  copper, 
it  will  give  the  smallest  copper  loss.  The  friction  and  windage  loss 
would  be  slightly  less  with  a  smaller  diameter,  while  with  a  greater  one 
the  power-factor  would  be  better  and  consequently  the  efficiency  would 
be  slightly  higher  on  account  of  the  slightly  smaller  current.  The  last 
two  considerations  about  offset  one  another,  and  hence  we  may  say, 
the  diameter  for  the  minimum  amount  of  copper  is  also  the  diameter 
for  the  maximum  efficiency. 

It  remains  to  consider  the  diameter  which  will  give  the  best  power- 
factor.  In  order  that  the  power-factor  may  be  high,  it  is  necessary 
that  the  leakage  factor  be  as  low  as  possible.  This  follows  from  the  fact 
that  the  maximum  value  of  the  power-factor  is  given  by  the  expression 

max.  cos  0=  --  .    We  have  also  derived  the  equation 

1  +  2(7 

-4 


132  THE  INDUCTION  MOTOR 

We  have  also  shown  that  o,  the  leakage  factor  can  be  represented  by 
the  expression 


It  must  be  kept  in  mind  that  this  expression  is  an  empirical  one,  derived 
from  experience,  although  the  general  form  might  have  been  predicted 
from  theory.  The  results  of  any  mathematical  transformations  will 
be  of  value,  only  as  the  original  formula  represents  the  facts.  In  the 
experience  of  the  author  as  well  as  in  that  of  others,  the  formula  has 
been  found  to  accord  well  with  the  results  obtained  from  the  completed 
machines. 

From  the  above  we  have, 

d      /       d  d 


Since  we  wish  to  study  the  effect  of  varying  /  and  b,  everything  else 
may  be  regarded  as  being  constant,  and  we  may  then  write: 


We    have  previously  shown  that  in  an  induction-motor  design    d~b 
const.      Since  I  is  proportional    to  d  we  may  write  instead  t2b  = 
Making  these  substitutions  we  have 


o=K\(3.62- 
Then  for  a  a  minimum, 


P  =  K2(i  .81  -0.9550). 
Substituting  the  value  of  K^  we  have: 


HIGH-  VS.  LOW-FREQUENCY  MOTORS  133 

in  which  p  is  the  pole  pitch  or  in  the  case  of  short-pitch  windings  the 
coil  pitch,  b  is  the  core  length,  and  SQ  is  the  percentage  opening  of  the 
slots.  Thus  we  have 

For  wide-open  slots,  / =0.866. 
For  half -open  slots  ,  t  =  1.336. 
For  closed  slots  ,  t=i.8ib. 

It  will  be  noted  that  to  obtain  the  maximum  power-factor,  we  require 
a  much  greater  pole  pitch,  or  what  is  equivalent,  a  much  greater  arma- 
ture diameter,  than  is  required  to  obtain  the  minimum  amount  of 
copper,  and  the  best  efficiency.  It  should  also  be  noted  that,  contrary 
to  a  rather  prevalent  belief,  an  increase  in  the  diameter  of  the  rotor 
does  not  always  result  in  an  improvement  in  the  power-factor. 


LENGTH  OF  AIR  GAP  FOR  BEST  POWER-FACTOR 
Writing  as  before, 

d  d 


and  changing  this  to  the  form 


we  see  at  once  that  this  will  be  a  minimum  for  d=o,  and  will  have  the 
value 


From  this  it  is  apparent ,  as  might  have  been  expected,  that  the  smaller 
the  air  gap,  the  better  will  be  the  power-factor  of  the  motor.  Of  course 
mechanical  considerations  will  deter  us  from  making  the  gap  as  short 
as  would  be  desirable.  The  practicable  limits  will  usually  lie  between 
the  values  0.02  in.  and  0.125  in.  The  larger  values  are  practicable 
only  with  motors  of  large  pole  pitch.  This  implies  in  general  motors 
for  low  frequency. 


134  THE  INDUCTION  MOTOR 

MODIFICATIONS  INTRODUCED  IN  PRACTICE 

In  practice,  the  application  of  the  above  principles  is  modified  by 
the  fact  that  it  is  desirable  to  use  as  few  different  diameters  as  possible- 
This  is  of  course  on  account  of  the  reduction  thereby  made  possible 
in  the  number  of  blanking  punches,  bearing  arms,  etc.  In  some  cases 
it  is  customary  to  build  three  different  sizes  of  motors  of  the  same  speed 
on  the  same  diameter  of  frame.  Thus  a  motor  of  10  h.p.  at  1200  rev. 
per  min.,  might  have  a  core  of  16  in.  diameter  and  4  in.  long.  The  15- 
and  20-h.p.  motors  of  the  same  speed  might  have  the  same  diameter, 
and  net  core  lengths  respectively  of  6  and  8  inches.  The  three  motors 
would  have  gradually  better  characteristics  as  they  increased  in  size. 
Thus  the  efficiency  would  be  better  in  the  larger  motors,  since  the  per- 
centage of  idle  copper  would  be  less.  The  percentage  loss  in  friction 
and  windage  would  likewise  be  less  in  the  larger  machines. 

The  leakage  factor  would  also  improve  as  the  core  length  increased, 
since  the  reactance  of  the  end  connectors  would  be  proportionately 
less.  Consequently  the  power-factor  and  pull-out  point  of  the  larger 
motors  would  be  somewhat  better. 

SIZE  OF  COPPER  CONDUCTORS 

The  determination  of  the  size  of  conductors  to  use  in  any  electrical 
machine  may  be  considered  from  two  standpoints.  The  conductor 
must  be  large  enough  to  keep  the  temperature  rise  within  a  certain  limit. 
They  should  be  large  enough  to  prevent  excessive  power  loss  for  the 
type  of  machine  in  question.  If  the  size  of  the  conductor  used  did  not 
influence  any  other  dimension  of  the  motor,  the  problem  of  determining 
the  proper  cross-section  for  given  conditions  would  be  comparatively 
simple.  This,  of  course,  rarely  happens.  The  following  can  be  con- 
sidered as  applying  only  to  a  case  of  the  kind  mentioned.  For  example, 
a  design  might  be  under  consideration  in  which  there  was  plenty  of 
room  in  the  slots  for  either  of  two  sizes  of  wire.  The  smaller  might  be 
satisfactory  as  far  as  heating  was  concerned,  and  the  question  would 
arise  whether  or  not  it  was  advisable  to  use  the  larger  wire  on  account 
of  the  better  efficiency  attainable. 

As  will  be  shown  presently,  the  loss  per  pound  of  copper  in  watts 
can  be  expressed  by  the  formula, 


where  Aa  denotes  circular  mils  per  ampere. 


HIGH-  VS.  LOW-FREQUENCY  MOTORS  135 

If  we  assume  that  the  motor  or  other  apparatus  is  used  an  average 
of  six  hours  per  day,  300  days  in  the  year,  and  that  the  value  of  the 
energy  wasted  is  $0.01  per  k.w.-hr.,  the  cost  of  the  energy  lost  per 
year  per  pound  of  copper  is, 

8        /7oo\2     70,600 
6X3ooXo.oiX X(-LT-]   =     A  o     dollars. 


If,  on  the  other  hand,  the  value  of  the  copper  installed  in  the  motor  be 
assumed  to  be  $0.25  per  pound,  and  the  interest  and  depreciation  be 
taken  as  10  per  cent,  we  have  a  fixed  charge  of  $0.025  Per  pound  of  copper 
per  year.  The  total  cost  of  operation  will  be  approximately  a  minimum 
when  the  interest  cost  is  equal  to  the  value  of  the  power  lost,  or, 


70,600 
Solving  this  we  get, 


0.025. 


4,  =  1680. 

Since  this  result  is  based  on  the  supposition  that  nothing  else  is 
affected  by  the  change  in  the  size  of  the  copper  conductors,  about  the  only 
practical  application  we  can  make  of  it  is  the  conclusion  that  the  con- 
ductor should,  considered  by  itself,  be  in  general  much  larger  than 
would  be  dictated  by  other  considerations.  If,  therefore,  we  could 
in  a  particular  case  use  a  wire  one  size  larger  than  would  ordinarily 
be  employed,  without  increasing  the  size  of  the  slots  or  altering  any 
other  dimensions,  it  would  pay  to  use  the  larger  wire.  Of  course 
even  this  limited  use  of  the  deduction  would  in  many  cases  be  prohibited 
by  commercial  conditions.  It  also  indicates  the  most  economical  size 
of  wire  to  use  in  the  leads  and  the  coil  connections.  If  a  motor  is  to 
be  used  a  less  number  of  hour  per  year  than  above  indicated,  a  smaller 
size  of  wire  would  be  preferable. 

TWO-PHASE  AND  THREE-PHASE  MOTORS  COMPARED 

In  every  respect,  the  two-phase  motor  is  slightly  inferior  to  the 
three-phase.  The  difference  is  slight,  it  is  true,  but  "it  unquestionably 
exists.  This  slight  inferiority  is  mainly  due  to  the  fact  that  in  the  three- 
phase  motor,  the  angular  breadth  of  one-phase  winding  is  60  degrees, 
while  in  the  two-phase  it  is  90  degrees.  This  results  in  the  breadth 
coefficient  in  the  three-phase  motor  being  better  than  in  the  two-phase. 


136  THE  INDUCTION  MOTOR 

As  was  shown  on  page  98,  these  values  are,  respectively,  0.953  an^ 
0.900.  As  a  consequence,  with  the  same  number  of  turns  and  the  same 
flux,  a  three-phase  motor  would  have,  at  its  terminals,  a  voltage  1.06 
times  as  great  as  that  of  the  two-phase  motor.  It  could  obviously 
carry  the  same  current,  and  consequently  the  input  and  the  output 
would  be  approximately  6  per  cent  greater  for  a  three-phase  motor 
than  for  the  same  core  wound  for  three-phase.  Of  course  in  practice 
the  two  motors  would  be  rated  the  same,  and  the  two-phase  motor  would 
in  consequence  operate  at  a  slightly  higher  temperature.  To  make 
up  for  the  lower  voltage  per  winding,  the  current  would  be  increased 
6  per  cent.  The  stator  copper  loss  would  therefore  be  increased  in  the 
proportion  of  i.o62  or  approximately  12  per  cent.  Hence  with  the  same 
output  the  losses  of  the  two-phase  motor  are  greater  than  those  of  the 
three-phase  machine,  and  its  efficiency  is  somewhat  lower. 

In  a  similar  manner  it  could  easily  be  shown  that  a  six-phase  motor 
would  be  slightly  better  than  a  three-phase  one,  and  so  on.  It  is  easy 
to  connect  transformers  so  as  to  change  a  three-phase  primary  to  a  six- 
phase  secondary.  In  fact  all  that  is  necessary  is  to  connect  the  prima- 
ries to  the  line  in  delta,  and  bring  out  all  six  secondary  leads  to  the 
motor.  The  neutral  points  of  the  three  secondary  windings  may  be 
connected  together  or  not  as  desired.  However,  the  complication  is 
greatly  increased  by  such  an  arrangement,  since  the  switches,  fuses, 
etc.,  all  would  have  to  be  six-pole  instead  of  three,  and  as  a  consequence 
the  arrangement  is  not  used  in  practice. 


DETERMINATION  OF  COPPER  LOSSES 

It  is  customary  to  express  the  size  of  wires  in  circular  mils,  a  circular 
mil  being  the  area  of  a  circle  o.ooi  in.  in  diameter.  The  circular  milage 
of  a  wire  is  therefore  equal  to  the  square  of  the  diameter  in  mils.  The 
most  convenient  way  of  estimating  the  size  of  wire  to  carry  a  given 
current,  is  to  allow  a  certain  number  of  circular  mils  per  ampere.  The 
number  to  allow  is  of  course  based  on  experience  with  similar  windings. 

This  also  leads  to  a  very  convenient  way  of  estimating  the  copper 
loss  in  a  given  winding.  This  is  based  on  the  fact  that  when  we  have 
such  a  current  that  the  circular  mils  per  ampere  are  700,  the  loss  in 
the  copper  is  eight  watts  per  pound.  This  figure  of  course  varies 
with  the  temperature,  and  the  above  is  correct  only  for  50  degrees  Centi- 
grade. The  loss  per  pound  will  obviously  be  proportional  to  the  square 
of  the  current,  or  inversely  proportional  to  the  square  of  the  circular  mils 


HIGH-  VS.  LOW-FREQUENCY  MOTORS  137 

per  ampere.     If  then  Aa  denotes  the  circular  mils  per  ampere,  and  G 
the  weight  <4  the  wire  in  pounds,  we  have  the  copper  loss 


This  formula  is  particularly  useful  in  determining  the  losses  in  squirrel- 
cage  rotors,  since  these  have  no  definite  resistance,  or  at  least  there  are 
no  definite  points  between  which  we  can  say  the  resistance  is  so  many 
ohms. 

IRON  LOSSES 

The  losses  in  the  iron  of  an  induction  motor  consist  of  two  parts: 
the  hysteresis  loss  and  the  eddy-current  loss.  The  former  is  propor- 
tional to  the  weight  of  the  iron,  and  the  frequency,  and  is  dependent 
upon  the  maximum  value  of  the  flux.  The  loss  varies  with  this  maximum 
value  in  a  more  or  less  irregular  manner,  depending  upon  the  composi- 
tion of  the  iron.  It  is  generally  assumed  that  it  is  proportional  to 
the  1.6  power  of  the  maximum  flux  density.  This  assumption  is  merely 
a  convenient  approximation,  but  accords  with  the  average  conditions 
as  well  as  any  other  known  value.  The  loss  due  to  hysteresis  may  then 
be  expressed  by  the  following  formula: 


in  which  y  is  a  constant,  depending  upon  the  quality  of  the  iron. 

Similarly  the  eddy-current  loss  is  proportional  to  the  weight  of 
the  iron,  to  the  square  of  the  frequency,  to  the  square  of  the  maximum 
flux  density,  and  inversely  proportional  to  the  square  of  the  thickness 
of  the  laminations.  The  eddy  current  loss  can  then  be  expressed  by 
the  following  formula: 


in  which  as  before  k  is   a  constant  depending  upon  the  conductivity  of 
the  iron. 

In  practice,  iron  for  use  in  electrical  machinery  is  tested  in  various 
ways.  One  of  the  simplest  of  these  is  to  make  the  sheets  to  be  tested 
the  core  of  a  transformer  having  a  single  winding.  The  winding 
is  permanent,  and  is  constructed  in  such  a  manner  as  to  leave  a  narrow 
but  long  opening.  The  uncut  sheets  are  then  inserted  in  this  opening 
and  bent  around  to  form  a  complete  magnetic  circuit.  If  the  wire 
is  of  reasonable  size,  the  loss  in  the  apparatus  when  an  alternating 


138  THE  INDUCTION  MOTOR 

current  of  the  usual  frequency  is  applied  to  it  is  practically  all  iron 
loss.  This  is  measured  by  a  wattmeter,  and  the  voltage  at  the  terminal 
is  read  at  the  same  time.  From  this  latter  and  the  dimensions  of  the 
sheets,  the  flux  density  is  readily  calculated.  The  weight  may  be 
easily  obtained,  and  from  these  data  the  loss  per  pound  or  per  cubic 
inch  can  be  at  once  computed. 

It  will  be  noted  that  the  loss  obtained  is  the  total  iron  loss  including 
both  the  hysteresis  and  the  eddy-current  loss.  The  same  thing  is  true 
with  most  of  the  other  methods.  This  is,  in  general,  rather  an  advantage, 
since  if  the  curves  are  prepared  for  the  frequencies  most  used  by  the 
designer,  he  can  at  once  obtain  the  loss  per  pound  from  the  curves 
without  computation.  If  the  individual  losses  are  desired,  they  can  be 
obtained  by  making  two  tests  at  different  frequencies. 

It  must  not  be  thought,  however,  that  the  losses  in  the  iron  as  above 
obtained  can  be  applied  without  change  to  the  computation  of  the  losses 
in  the  completed  motor.  The  losses  in  the  actual  machine  will  be 
found  to  be  from  three  to  five  tunes  as  large  as  would  be  expected  from 
the  application  of  the  values  so  obtained.  Several  factors  contribute 
to  this  discrepancy. 

The  most  important  of  these  is  perhaps  the  fact  that  the  flux  is 
not  uniformly  distributed  throughout  the  metal  of  the  laminations. 
It  tends  to  take  the  shortest  path,  and  this  results  in  the  flux  density 
being  greater  in  some  parts  of  the  iron  than  in  others.  .It  might  appear 
at  first  sight  that  since  the  flux  is  less  in  other  paths,  the  average  loss 
would  be  the  same.  That  this  is  not  so  appears  at  once  when  we 
consider  that  the  hysteresis  loss  is  proportional  to  the  1.6  power  of  the 
flux  density,  and  the  eddy-current  loss  to  the  square.  This  results 
in  the  uneven  distribution,  giving  a  much  larger  loss  than  would  result 
from  a  uniform  distribution. 

Another  factor  tending  to  increase  the  losses  is  the  filing  of  the  slots. 
It  is  generally  considered  necessary  to  do  this  in  order  that  the  sharp 
edges  may  not  cut  the  insulation,  and  that  the  coils  may  be  easily  inserted 
in  the  slots.  The  filing  causes  a  conducting  film  to  be  formed  on  the 
inside  of  the  slots,  and  this  results  in  increased  eddy-current  losses. 

In  computing  the  iron  losses,  those  in  the  teeth  and  those  in  the 
body  of  the  iron  should  for  the  greatest  accuracy  be  computed  separately, 
since  in  general  the  densities  are  not  the  same.  In  practice,  however, 
this  is  frequently  not  done,  and  an  average  value  is  taken  for  the  whole 
motor.  One  method  of  doing  this  is  to  compute  the  whole  weight  of 
the  stator  as  though  it  were  solid,  i.e.,  making  no  allowance  for  the 


HIGH-  VS.  LOW-FREQUENCY  MOTORS 


139 


slots,  and  multiply  by  the  factor  corresponding  to  the  flux  density 
in  the  body  of  the  iron.  This  procedure  is  allowable  since  the  computa- 
tion of  this  loss  is  in  any  case 
subject  to  large  errors  on 
account  of  unavoidable  varia- 
tions in  the  iron,  and  since  12 
the  extra  weight  taken  com-  10 
pensates  roughly  for  the  extra  8 
loss  in  the  teeth.  Fig.  66  e 
gives  average  values  for  the  4 
iron  loss  per  pound  computed  „ 
as  above  indicated.  The  curves 
were  obtained  from  actual  tests 
of  completed  motors.  The  iron 
was  of  good  commercial  quality. 


4 

/* 

ttCy 

cles 

i-i 

h 

/ 

a 

/ 

I 

/ 

z> 

Cyc 

les^ 

/ 

^, 

^ 

^-*' 

^ 

—  - 

—  • 

—  " 

-kii 

,_u 

*SJ 

x^ 

*L] 

n. 

0     10    30    30    40    50    60    70    80    90 

FIG.  66. — Hysteresis  and  Eddy  Current 

Loss  in  Sheet  Iron. 


ROTOR  IRON  Loss 

In  general  it  is  assumed  that  the  iron  loss  in  the  rotor  of  an  induc- 
tion motor  operating  near  synchronism  is  zero.  This  is  allowable  since 
on  account  of  the  low  frequency  in  the  rotor  the  eddy-current  loss 
almost  disappears  and  the  hysteresis  loss  does  not  exceed  a  few  per 
cent  of  its  value  at  the  applied  frequency.  Of  course  this  does  not  apply 
in  the  case  of  motors  arranged  to  run  at  speeds  far  from  syn- 
chronism, as  in  the  case  of  the  primary  motor  of  two  used  in 
concatenation. 

In  certain  cases  however  this  assumption  is  far  from  true.  This  can 
perhaps  best  be  shown  by  an  actual  example.  Thus  in  Fig.  67  are 
represented  the  stator  and  rotor  slots  of  a  certain  motor,  as  originally 
built.  With  these  proportions  the  iron  loss  was  about  6  k.w.  This 
loss  was  about  three  times  the  expected  value  and  the  motor  had  a  rise 
in  temperature  under  full  load  of  about  80  degrees  Centigrade.  This 
led  to  an  examination  of  the  design,  and  it  was  at  once  seen  that  the 
relation  of  the  stator  and  rotor  slots  was  such  that  a  tooth  in  the  rotor, 
such  as  A,  would  be  at  times  so  situated  that  almost  all  the  flux  from  a 
stator  tooth  would  pass  through  it,  and  at  other  times,  when  situated 
in  the  position  of  tooth  B,  there  will  be  very  little  flux  passing  through 
it.  It  will  be  apparent  at  once  that  this  construction  will  give  rise  to 
a  high-frequency  pulsation  of  the  flux  in  the  rotor  teeth  and  a  similar 
fluctuation  in  the  stator  teeth.  The  complete  period  of  this  pulsation 


140 


THE  INDUCTION  MOTOR 


will  be  the  time  occupied  by  a  rotor  tooth  in  passing  from  one  stator 
tooth  to  the  next.  The  machine  in  question  had  eight  poles,  54  stator 
slots  and  115  rotor  slots.  Since  it  was  operated  on  25-cycle  current, 


[Diam.9 
FIG.  67 —Stator  and  Rotor  Slots. 

the  frequency  of  the  flux  in  the  stator  was  25,  that  in  the  body  of  the 
rotor  at  4  per  cent  slip,  was  one  cycle  per  second.     The  frequency  of 


Stator  54  Slots 
Rotor  79  Slots 


Diam.  9 
FIG.  68. — Stator  and  Rotor  Slots. 


the  pulsation  in  the  rotor  teeth  was,  however,  54X500-7-60  =  450  in  the 
rotor  and  115X500-7-60  =  960  in  the  stator.  It  will  be  readily  seen 
that  even  a  comparatively  small  fluctuation  of  the  flux  at  such  a  high 
frequency  will  give  rise  to  considerable  losses.  This  is  particularly  true  of 


HIGH-  VS.  LOW-FREQUENCY  MOTORS  141 

the  eddy-current  component,  both  on  account  of  the  high  frequency  and 
on  account  of  the  filing  of  the  slots. 

It  is  of  course  true  that  a  large  proportion  of  the  apparent  fluctua- 
tion is  prevented  by  the  production  of  currents  in  the  rotor  bars. 
As  soon  as  the  flux  through  a  tooth  starts  to  change,  a  current  is  set  up 
in  the  rotor  bars  surrounding  the  tooth  in  question,  and  this  current 
tends  powerfully  to  prevent  the  change  of  flux.  This  is  true  to  any 
great  extent,  only  in  the  case  of  a  squirrel-cage  rotor.  It  is  perhaps 
impracticable  to  calculate  the  magnitude  of  this  shielding,  and  in  any 
event  the  lessened  iron  loss  due  to  the  shielding  is  in  large  measure  com- 
pensated for  by  the  increased  copper  loss  in  the  rotor. 

The  remedy  adopted  in  this  case  was  to  change  the  number  of  rotor 
slots  to  79,  adopting  at  the  same  time  a  more  nearly  closed  slot  as  shown 
in  Fig.  68.  It  will  be  apparent  at  once  that  the  fluctuation  will  be 
greatly  reduced.  The  test  bore  out  this  conclusion,  as  the  iron  loss  after 
the  change  was  only  2.2  k.w.  instead  of  6  k.w.* 

ESTIMATION  OF  HEATING 

In  designing  induction  motors  for  general  use,  the  usual  require- 
ment as  regards  heating  is  that  the  rise  in  temperature  under  full 
normal  load  for  an  indefinite  time  shall  not  exceed  40  degrees  Centi- 
grade. To  this  is  usually  added  the  further  requirement  that  under 
a  load  of  125  per  cent  of  normal  for  two  hours  immediately  following 
the  full -load  run,  the  rise  shall  not  exceed  55  degrees  Centigrade.  Both 
the  above  are  for  a  room  temperature  of  25  degrees,  and  are  to  be  cor- 
rected one-half  of  one  per  cent  for  every  degree  that  the  temperature 
differs  from  25  degrees. 

In  estimating  the  rise  of  a  given  machine,  or  in  designing  a  machine 
so  as  not  to  exceed  a  certain  rise,  there  are  several  different  methods 
open  to  the  designer.  In  the  first  place,  such  values  are  chosen  for  the 
circular  mils  per  ampere,  the  flux  density  in  the  iron,  etc.,  that  the 
probability  is  that  the  temperature  rise  will  be  conservative.  Thus 
in  the  average  6o-cycle  machine,  if  the  circular  mils  per  ampere  are  not 
less  than  600,  and  if  the  flux  density  in  the  stator  does  not  exceed  about 
40,000  lines  per  square  inch,  depending  upon  the  quality  of  the  iron, 
and  if  the  ventilation  is  reasonably  good,  the  probability  is  that  the 
machine  will  not  overheat. 

*  For  a  full  discussion  of  iron  losses,  see  "  Calculation  of  Iron  Losses  in  Dynamo 
Electric  Machinery,"  by  I.  E.  Hansen,  Transactions  of  A.  I.  E.  E.,  Vol.  XXVIII, 
Part  II.  p.  993. 


142  THE  INDUCTION  MOTOR 

The  heating  is  of  course  greatly  affected  by  the  peripheral  speed  of 
the  rotor,  the  number  and  size  of  the  ventilating  ducts,  the  size  and  pro- 
portions of  the  machine,  etc.  The  above  is  therefore  valuable  merely 
as  a  guide,  and  may  be  accepted  as  conclusive  only  in  case  the  machine 
is  similar  to  another  in  design,  and  this  one  has  proven  satisfactory  in 
this  respect.  Thus  if  a  2o-h.p.  motor  had  a  core  length  of  10  ins., 
it  is  certain  that  a  i5-h.p.  motor,  having  the  same  flux  and  current  den- 
sity, and  having  a  core  length  of  7^  ins.,  would  operate  at  a  slightly  lower 
temperature.  The  fact  that  the  temperature  would  be  lower  is  due 
to  the  greater  radiating  surface  in  the  second  motor,  in  proportion  to  its 
losses.  This  fact  is  useful  in  that  it  allows  us  to  dispense  with  the 
labor  of  computing  the  heating  of  all  of  a  line  of  motors.  If  all  are 
designed  with  the  same  constants,  it  is  only  necessary  to  calculate  the 
heating  of  one  motor  of  each  diameter.  The  computation  is  of  course 
to  be  performed  for  the  motor  of  the  greatest  core  length. 

The  opinions  of  different  designers  as  to  the  best  method  of  estimating 
the  heating  differ  very  widely.  All  of  these  methods  are,  however, 
based  on  the  assumption  that  the  rise  in  temperature  is  proportional 
to  the  number  of  watts  radiated  per  square  inch.  The  number  to  be 
allowed  varies  greatly  with  the  peripheral  velocity  of  the  rotor,  the 
dimensions  of  the  machine,  etc.  The  uncertainty  and  lack  of  agree- 
ment as  to  method  arise  mainly  on  account  of  the  difficulty  of  estimating 
the  actual  radiating  surface  and  the  number  of  watts  to  be  allowed  per 
square  inch.  Thus  some  designers  count  as  the  radiating  surface  the 
so-called  barrel  surface  of  the  machine,  i.e.,  the  interior  area  of  the  stator 
surface  including  the  projecting  part  of  the  coils.  On  this  assumption, 
from  two  to  ten  watts  per  square  inch  can  be  allowed,  depending  . 
upon  the  peripheral  speed  of  the  rotor. 

This  method  has  not  given  good  results  in  the  experience  of  the 
author.  In  his  opinion,  a  preferable  method  is  to  compute  the  total 
radiating  surface  of  the  machine.  It  is  of  course  a  matter  of  judgment 
whether  or  not  all  of  a  given  surface,  as  the  gap  space  or  the  interior  of 
the  air  ducts,  should  be  counted.  It  is  his  practice  in  general  to  count 
all  of  the  exterior  of  the  stator,  the  interior  of  the  rotor,  the  ends  of 
both  the  stator  and  the  rotor,  and  one  side  only  of  the  air  gap  surface 
and  of  the  air  ducts.  The  accompanying  curve,  Fig.  69,  is  the  result 
of  a  large  number  of  tests  upon  completed  motors.  The  radiating 
surface  was  computed  in  the  manner  indicated.  The  losses  considered 
were  all  the  losses  in  the  motor  except  that  in  bearing  friction  and  the 
windage,  and  are  the  losses  that  will  give  a  rise  of  40  degrees  Centigrade. 


HIGH-  VS.  LOW-FREQUENCY  MOTORS 


143 


Curve  A  is  for  motors  having  plain  rotors,  and  curve  B  for  those  having 
small  wings.  The  values  plotted  were  taken  from  a  wide  range  of 
motors,  and  will  be  found  to  give  reasonably  consistent  results  in 
practice. 

Where,  however,  a  large  number  or  motors  have  been  built  in  the 
same  frame,  the  author  prefers  another  method.  It  is  his  belief,  based 
upon  a  large  number  of  tests,  that  in  the  case  of  induction  motors, 
having  frames  of  the  conventional  pattern,  the  heat-dissipating  power  of 
the  machine  depends  almost  entirely  upon  the  dimensions  of  the  frame, 
and  upon  the  peripheral  speed  of  the  rotor.  This  seems  reasonable, 
since  the  tendency  of  the  manufacture  is  of  course  to  use  the  smallest 


1234 
FIG.  69. — Allowable  Watts  per  Square  Inch  of  Core  Surface. 

possible  frame,  and  in  consequence  the  radiating  surface  of  the  active 
iron  of  the  motor  will  not  vary  much  in  different  machines  in  the  same 
frame.  In  any  case  it  cannot  be  doubted  that  the  radiating  surface 
of  the  frame  itself  is  of  the  greatest  assistance  in  keeping  the  motor 
cool.  This  method  has  been  found  to  give  much  more  consistent 
results  than  any  other  that  has  been  tried,  and  has  the  further  merit  of 
being  very  simple  to  apply.  Each  new  motor  in  a  given  frame  is  care- 
fully tested  for  losses  and  the  rise  in  temperature  under  full  load  for  an 
indefinite  time  taken.  The  actual  watts  lost  in  the  motor  are  then 
multiplied  by  40  divided  by  the  actual  rise,  and  the  result  taken  as  the 
allowable  total  watts  for  the  frame,  at  the  given  peripheral  speed, 
without  exceeding  a  rise  of  40  degrees  Centigrade.  The  number  of 
watts  that  can  be  radiated  without  exceeding  the  allowable  rise  are 


144  THE  INDUCTION  MOTOR 

plotted,  with  the  peripheral  speeds,  and  as  soon  as  a  sufficient  number 
of  observations  are  available,  the  curve  representing  the  average  of 
these  points  is  drawn  in.  Used  with  discretion,  this  will  give  a  reliable 
guide  for  all  future  designs  in  the  same  frame. 

As  a  guide  in  case  only  one  observation  is  available  on  a  given  frame, 
the  author  has  found  that  the  allowable  watts  for  the  frame  can  be 
represented  by  the  following  formula: 

P  =  P0(i  +  o.oooi65), 

where  P  represents  the  allowable  power  in  watts  for  the  frame  at  the 
peripheral  speed  S,  expressed  in  feet  per  minute,  and  PQ  represents 
the  allowable  power  in  watts  at  standstill.  The  above  is  for  machines 
only  reasonably  well  ventilated  and  without  wings  on  the  rotor.  In 
case  of  very  good  ventilation  and  a  rotor  provided  with  good-sized  wings, 
the  formula  will  be: 


An  attempt  has  also  been  made  to  apply  the  formula  to  frames  on 
which  no  tests  are  available.  In  this  case  the  watts  that  can  be  dissipated 
are  given  approximately  by  the  formula: 

P  =  (i  .3  +  0.00022^)^4     (without  wings), 
and 

P=  (1.3  +  0.00044^)^4     (with  wings). 

The  surface  A  to  be  taken  is  that  of  the  total  surface  of  a  cylinder 
which  would  just  cover  the  main  part  of  the  frame,  the  ends  of  the 
cylinder  cutting  through  about  the  middle  of  the  bearings.  Such  a 
formula  can  in  the  nature  of  the  case  be  nothing  more  than  a  rough 
approximation,  and  wherever  it  is  possible,  the  results  of  a  test  of  the 
frame  to  be  used,  or  of  a  similar  one  of  nearly  the  same  size  should  be 
used.  The  writer  would  place  more  reliance  upon  the  computation 
of  the  watt  per  square  inch  of  core  and  would  use  the  above  principally 
as  a  check. 


CHAPTER  X 
FRACTIONAL-PITCH  WINDINGS 

THE  proper  pitch  to  use  in  the  coils  of  an  induction  motor  is  of 
great  importance.  In  most  other  classes  of  electrical  machinery  full- 
pitch  windings  are  the  rule.  In  the  case  of  induction  motors,  however, 
the  short-pitch  winding  is  the  rule  and  the  full-pitch  one  the  exception. 

In  brief,  the  advantage  of  the  fractional-pitch  winding  is  that  the 
amount  of  wire  in  the  end  connections  is  reduced,  and  consequently 
the  inductance  and  resistance  of  the  stator  or  rotor,  and  the  coils  are 
distributed  in  more  slots  per  phase  per  pole,  giving  essentially  the  same 
effect  as  though  the  number  of  slots  were  increased.  The  disadvantage 


FIG.  70—  Full  Pitch  Coils. 

is  that  a  coil  does  not  enclose  all  the  flux  from  a  pole  and  consequently 
the  number  of  turns  must  be  increased  or  a  greater  flux  density  employed. 
We  will  now  consider  these  points  more  in  detail. 

In  Fig.  70  is  shown  a  section  of  a  stator  wound  with  a  full-pitch 
winding.  Fig.  71  shows  a  similar  stator  wound  with  a  fractional- 
pitch  winding.  In  both  cases  the  winding  is  for  tnree  phase,  and  the 
number  of  slots  is  six  per  pole,  or  two  per  phase  per  pole.  In  Fig.  70 
the  pitch  of  the  coils  is  i  to  7,  2  to  8,  etc.  In  Fig.  71  it  is  i  to  6,  2  to  7, 
etc.  The  winding  is  such  that  there  are  as  many  coils  as  slots.  Placing 
the  coils  in  the  slots  with  the  pitches  given,  it  will  be  readily  seen  that 
in  the  case  of  the  full-pitch  winding  a  slot  will  contain  only  coils  of  a 
certain  phase.  Thus,  in  Fig.  70,  slots  i  and  2  contain  coils  of  the  A 
phase  only.  Slots  3  and  4  those  of  the  B  phase,  and  4  and  5  those  of 
the  C  phase. 

i4S 


146  THE  INDUCTION  MOTOR 

In  the  case  of  the  short-pitch  winding,  however,  it  will  be  seen 
that  this  is  not  the  case.  Thus  slot  i  contains  only  A  coils,  but  slot 
2  has  one  A  coil  and  one  B  coil.  If  we  consider  the  coils  belonging  to 
any  one  phase,  say  A,  it  will  be  found  that  the  coils  of  this  phase  are 
distributed  one  in  slot  6,  two  in  slot  7,  and  one  in  slot  8.  It  is  evident 
that  this  will  affect  the  motor  to  practically  the  same  extent  as  though 
the  motor  were  actually  provided  with  three  slots  per  phase  per  pole 
instead  of  with  two,  and  the  calculations  are  usually  made  as  though 
this  were  the  case. 

If,  instead  of  making  the  pitch  in  Fig.  71  i  to  6,  it  had  been  made 
i  to  5,  it  will  be  readily  seen  that  the  windings  of  each  phase  would 
have  been  distributed  in  four  slots  per  phase  per  pole.  If  an  attempt 
were  made  to  extend  this  further,  it  is  apparent  that  a  slot  would  be 
left  between  the  two  parts  of  the  winding  of  a  given  phase,  and  no 


3156 

FIG.  71.— Fractional  Pitch  Coils. 

advantage  would  be  gained  as  far  as  the  distribution  of  the  winding  is 
concerned. 

The  number  of  slots  per  phase  per  pole  actually  occupied  by  the 
windings  of  a  given  phase  is  sometimes  called  the  equivalent  slots 
per  phase  per  pole.  To  determine  this  number,  we  divide  the  total 
number  of  slots  by  the  number  of  poles  and  by  the  number  of  phases, 
and  add  to  the  quotient  the  number  of  slots  by  which  the  coils  fall  short 
of  being  full  pitch. 

That  the  amount  of  idle  copper  is  reduced  by  using  short-pitch 
windings  is  self-evident.  The  idle  copper  per  coil,  other  things  being 
equal,  will  be  proportional  to  the  coil  pitch.  As  will  be  shown  presently, 
the  number  of  turns  must  be  increased  if  the  flux  density  is  not 
increased.  This  of  course  tends  to  offset  the  advantage  gained  by 
the  shorter  length  of  end  connections.  However,  the  increase  in 
turns  for  a  moderate  shortening  in  the  coil  pitch  is  very  slight,  while 
the  saving  in  end  copper  is  considerable.  The  determination  of  the 
pitch  which  gives  the  minimum  amount  of  copper  will  be  given  later. 


FRACTIONAL-PITCH  WINDINGS 


147 


The  calculation  of  the  reduction  in  voltage  per  coil  when  short- 
pitch  winding  is  used  is  very  simple.  In  Fig.  72  let  E  and  E  repre- 
sent the  e.m.fs.  generated  in  the  two  sides  of  a  coil.  The  flux  wave 
is  assumed  to  be  of  the  sine  shape,  so  that  the  e.m.f.  waves  are  also 
sine  waves,  and  can  consequently  be  represented  by  vectors.  The 
angle  0  represents  the  electrical  angle  between  the  two  sides  of  the 
coil.  For  a  full-pitch  winding,  the  angle  0  would  of  course  be  180°. 


FIG.  72. — E.M.F.  Relations  in  Fractional  Pitch  Coils. 


In  the  winding  of  Fig.  71,  for  example,  since  only  five  instead  of  six 
slots  are  spanned  by  the  winding,  the  angle  would  be  f  of  180°  or  150°. 
In  Fig.  72  line  E\  is  the  resultant  of  the  two  e.m.fs.  E  and  E.  It 
will  be  readily  seen  that  its  value  is  equal  to  2E  sin  0/2.  Since  if  the 
winding  had  been  full  pitch  the  resultant  would  have  been  2E,  the  back 
e.m.f.  has  been  reduced  by  the  use  of  the  short-pitch  winding  in  the 
ratio  of  sin  0/2  to  one,  or  in  an  induction  motor,  if  the  flux  density  is 
to  be  kept  the  same,  the  turns  per  coil  must  be  increased  in  the 
same  proportion.  The  factor  for  the  complete  winding  is  of  course 
the  same  as  that  for  a  single  coil. 

It  is  of  interest  to  find  the  pitch  which  will  in  any  case  require 
the  minimum  amount  of  copper. 
In  Fig.  '73  let  /  represent  the 
length  of  the  straight  portions  of 
the  coils,  let  P  represent  the  pole 
pitch,  and  p  the  coil  pitch.  Also, 
let  Np  represent  the  number  of 
conductors  which  would  be  re- 
quired if  the  winding  were  of  full 
pitch  and  Np  the  number  actually 
required  with  the  shorter  pitch. 
If  then  k  represent  the  ratio  p/P, 
we  have: 


FIG.  73. — Relative  Length  of   Copper 
for  Full  Pitch  and  Short  Pitch  Coils. 


148  THE  INDUCTION  MOTOR 

The  length  of  one  conductor  will  be  L  =  l+i.$p  =  l+i.$kP,  if  we 
make  the  assumption  that  the  length  of  the  end  connection  is  i \  times 
the  coil  pitch.  This  is  a  fair  average  of  the  ratio  attained  in  practice. 
As  we  have  just  shown,  the  number  of  conductors  must  be  increased 
as  the  pitch  is  made  shorter,  in  such  a  proportion  that 


sn  -  =        sn 


The  total  length  of  conductor  required  is  then 
NP 


5inK) 


The  only  variables  in  this  equation  are  C,  the  total  length  of  copper, 
and  k,  the  ratio  of  the  actual  pitch  to  the  full  pitch.     To  determine 
what  value  of  k  gives  the  minimum  amount  of   copper,     we   must 
differentiate  C  with  respect  to  k  and  place  the  result  equal  to  zero. 
Thus 

dC  NP  {          7T  /7T     \  /7T     \ 

—  =0=  --  T—\*     -/-cos    -£    +1.5  sin  (-k) 

dk  .    2/7l     \  2  \2      I  \2      / 

sin2  (  —  k  ) 


This  readily  reduces  to 


tan  (-&)  =  i.045-  +  -£. 

P        2 


Expanding  tanf  —  k  J  we  have: 

-*+-(-) 

2  3\2/ 


, 

— +-  —     +--  — 1  +...  =1.045-+ -k. 

32/          I5\2/          3I5\2/  P         2 

Reducing  the  coefficients  of  k  it  will  be  found  that  they  are  all 
(with  the  exception  of  the  first)  very  nearly  equal  to  1.29.  Assuming 
that  they  are  constant  we  get 

...  =0.813-^. 


FRACTIONAL-PITCH  WINDINGS  149 

( 
This  series  may  be  replaced  by —^  giving  as  the  final  form 

1.23^  _  / 
l^k?  =P' 

The  ratio  l/P  is  fixed  from  the  construction  of  the  motor,   and 
substituting  its  value,  we  can  compute  the  value  of  k.     Since,  how- 


FIG.  74. — Curve  of  Values  of  "  k. 


ever,  the  equation  is  of  the  third  degree  in  k,  this  is  somewhat  difficult. 
In  Fig.  74,  however,  are  plotted  values  of  l/P  and  p/P  or  k.  The 
most  economical  pitch  can  at  once  be  found  from  the  curve.  Thus 
if  the  pole  pitch  were  10  ins.,  the  core  length  5  ins.,  and  the  coils  project 
i  in.  on  each  side  before  starting  to  bend,  the  ratio  l/P  is  0.7  and 
the  pitch  should  be  68  per  cent  of  full  pitch.  If  the  core  had  been  twice 
as  long,  or  10  ins.,  the  most  economical  pitch  would  have  been  74  per 
cent. 

In  applying  the  above,  one  should  not  lose  sight  of  the  fact  that 
although  the  use  of  the  pitch  recommended  will  lead  to  the  employ- 
ment of  the  minimum  amount  of  copper,  the  amount  of  copper,  and 


150  THE  INDUCTION  MOTOR 

the  number  of  wires  in  each  slot  is  increased.  This  may  lead  to  difficulty 
on  account  of  the  fact  that  the  slot  may  not  be  large  enough  to  hold 
the  wire,  or  if  a  deeper  slot  is  used,  the  depth  of  the  iron  back  of  the 
slots  may  have  to  be  increased,  and  thus  the  greater  amount  of  iron 
and  the  larger  frame  may  offset  the  saving  in  copper.  In  such  cases 
it  is  obviously  wise  to  use  a  coil  of  slightly  greater  pitch,  say  of  one 
tooth  more  than  indicated. 

The  above  demonstration  applies  equally  well  to  the  coils  of  an 
alternator  or  of  a  direct-current  machine,  provided  the  flux  follows 
a  sine  distribution.  As  a  matter  of  fact,  the  curves  of  flux  distribution 
of  such  machines  are  usually  flat  topped,  with  frequently  a  considerable 
interval  of  zero  flux.  In  this  case,  the  gain  by  using  short  pitch  coils 
is  even  greater  than  indicated  above.  In  fact  the  pitch  may  be 
shortened  to  almost  the  width  of  the  poles,  without  any  appreciable 
loss  in  generated  e.m.fs.  In  direct-current  machines,  however,  short- 
pitch  windings  are  not  much  used,  since  they  virtually  lessen  the 
neutral  zone. 

EFFECT  OF  SHORT  PITCH  UPON  STARTING  TORQUE 

As  has  been  previously  shown,  when  the  rotor  is  operating  near 
synchronism,  the  rotor  currents  have  a  powerful  effect  upon  the 
stator  magnetism  tending  to  cause  the  wave  of  flux  to  be  of  the  sine 
shape  and  to  rotate  at  uniform  velocity,  provided  the  applied  e.m.f. 
is  sinusoidal.  At  starting,  however,  since  the  rotor  is  at  rest,  this 
action  is  very  much  weaker.  This  is  the  case  since  the  frequency 
of  the  current  in  the  rotor,  being  the  full  line  frequency,  the  setting 
up  of  the  corrective  currents  is  very  much  hindered.  The  result  is 
that  in  both  the  squirrel-cage  type,  and  in  the  wound-rotor  type,  the 
wave  of  revolving  flux-  differs  materially  from  the  sine  shape.  The 
wave  is  not  only  distorted,  but  it  also  changes  its  shape  from  point 
to  point.  The  distortion  of  the  wave  would  in  itself  not  lessen  the 
starting  torque,  but  the  fluctuation  of  the  flux  does  seriously  reduce  it. 

The  use  of  a  short-pitch  winding  greatly  reduces  the  distortion 
of  the  flux  wave  during  the  starting  period,  and  hence  considerably 
increases  the  starting  torque.  That  this  will  be  so  can  be  readily 
seen,  since  in  the  short-pitch  winding  the  phases  overlap  each  other 
and  consequently  take  off  the  corners  of  the  flux  wave.  In  fact,  in 
the  original  United  States  patent  on  the  short-pitch  winding  (issued 
to  B.  J.  Lamme),  it  is  this  feature  which  is  emphasized  as  being  the 
principal  advantage  of  the  winding. 


FRACTIONAL-PITCH  WINDINGS 


151 


STATOR  AND  ROTOR  WINDINGS 

The  subject  of  windings  is  a  very  extensive  one,  and  to  treat 
it  in  a  comprehensive  manner  would  require  more  time  than  can 
possibly  be  given  to  it  here.  The  writer  will  therefore  merely 
attempt  to  point  out  some  of  the  distinctive  features  of  the  various 
windings,  and  would  refer  the  student  to  any  one  of  several  excellent 
works  that  have  appeared  on  this  subject  for  further  information. 

Any  direct  current  winding  might  be  used  on  an  induction  motor. 
As  a  matter  of  fact,  such  windings  are  rarely  if  ever  used,  except  in  the 
case  of  a  rotor  provided  with  a  commutator.  The  reason  for  this  is 


FIG.  75. — Comparison  of  Three-phase 
and  Six-phase  Windings. 


FIG.  76.— Six-phase  Winding. 


that  such  a  winding  would,  for  a  given  flux  density  and  a  given  number 
of  turns,  require  a  lower  e.m.f.  at  the  terminals  than  would  a  winding 
of  a  different  type,  and  in  consequence  the  rating  of  the  machine  would 
be  less.  The  reason  for  this  will  be  readily  seen  from  Figs.  75  and 
76.  If  the  number  of  conductors  is  large,  the  various  e.m.fs.  may  be 
regarded  as  being  short  segments  of  a  circle.  Using  the  direct-cur- 
rent winding  represented  in  Fig.  75  taps  would  be  taken  off  from  three 
points  120  electrical  degrees  apart  as  shown  at  A,  B,  and  C  for  a 
three-phase  winding.  The  counter  e.m.f.  developed  by  a  given  flux 
would  be  proportional  to  the  chord  of  the  circle  AB.  If  instead  of 
connecting  in  this  way,  we  break  the  connection  at  D  as  shown  in 
Fig.  76,  and  connect  across  the  circle,  the  voltage  developed  by  each 
phase  for  the  same  flux  as  before  will  be  proportional  to  twice  the 


152  THE  INDl/CTION  MOTOR 

chord  AD.  This  is  obviously  greater  than  AB  in  the  ratio  of  4  sin  30° 
to  2  sin  60°  or  1.152.  The  winding  shown  in  Fig.  76  will  therefore 
give  an  output  of  approximately  15  per  cent  more  than  would  a  direct- 
current  winding,  used  for  a  three-phase  motor. 

.  A  little  consideration  will  show  that  the  winding  of  Fig.  75  is  a 
true  three-phase  winding,  while  that  of  Fig.  76  is  in  reality  a  six-phase 
one.  On  account  of  the  interconnection  of  the  coils,  only  a  three-phase 
e.m.f.  appears  at  the  terminals.  The  same  principle  applies  to  the  case 
of  a  two-phase  winding.  For  the  best  results  the  winding  should  be 
a  four-phase  one,  and  the  connections  so  made  that  the  terminal  e.m.f. 
is  two-phase. 

STAR  OR  DELTA  CONNECTION 

It  is  obvious  that  the  windings  of  Fig.  76  might  be  joined  in  either 
a  star  or  a  delta  connection.  Either  will  work  with  entire  satisfaction 
and  the  choice  between  the  two  is  largely  a  matter  of  convenience. 
Other  things  being  equal,  however,  the  star  connection  should  be 
chosen.  This  is  on  account  of  the  fact  that  with  the  star  winding, 
since  only  57.7  per  cent  of  the  total  e.m.f.  is  applied  to  each  winding, 
the  number  of  turns  will  be  less  than  in  the  delta  connection.  The 
size  of  the  wire  must  of  course  be  greater  in  the  same  proportion, 
and  consequently  there  is  no  saving  in  copper,  but  the  fact  that  the 
labor  of  winding,  at  least  with  small  sizes  of  wire  is  less,  and  the  fact 
that  the  amount  of  room  required  for  insulation  is  less,  cause  the 
star  winding  to  be  preferred.  This  is  particularly  the  case  in  high- 
voltage  motors. 

In  the  case  of  large  low-voltage  machines,  the  designer  is  often 
seriously  restricted  in  his  choice.  Thus  it  might  readily  be  the  case 
that  with  a  star  winding,  in  order  to  obtain  the  desired  flux  density, 
1.7  turns  per  coil  would  be  required.  The  use  of  either  one  or  two 
turns  would  be  out  of  the  question,  the  one  giving  too  high  and  the 
other  too  low  a  flux  density.  If,  however,  a  delta  winding  be  used, 
the  turns  per  coil  should  be  increased  in  the  ratio  \/3-  Using  this 
winding  then  the  preferable  number  of  turns  would  be  2.94  and  three 
turns  would  be  entirely  satisfactory. 

POLE  CONNECTIONS 

It  is  of  considerable  advantage  to  a  manufacturer  to  be  able 
to  change  the  voltage  for  which  a  machine  is  wound,  without 
substituting  new  coils  and  reconnecting  the  entire  machine.  The 


FRACTIONAL-PITCH  WINDINGS  153 

ability  to  do  this  readily  may  also  be  of  the  greatest  value  after  the 
machine  is  in  the  hands  of  the  agent  or  of  the  customer. 

The  voltages  which  are  standard  for  induction  motors  are  no, 
220,  440,  and  550.  To  secure  the  above  property,  the  machine  would 
be  designed  primarily  for  440  volts,  and  all  of  the  stator  windings 
would  be  in  series.  To  reconnect  for  220  volts,  it  would  be  necessary 
merely  to  divide  the  windings  into  groups  corresponding  to  the 
various  poles.  All  of  these  groups  would  be  alike,  with  any  of  the 
common  windings.  The  groups  would  then  be  connected  together 
so  that  half  of  the  groups  corresponding  to  any  phase  would  be  in  series, 
and  the  two  halves  would  then  be  connected  in  parallel.  In  case  the 
number  of  poles  is  divisible  by  four,  the  process  can  be  extended  to 
the  case  of  a  no-volt  winding,  by  connecting  one -fourth  of  the  coils 
in  series  and  the  four  series  in  parallel.  For  550  volts,  a  special  winding 
is  usually  required. 

In  the  case  of  large  low- voltage  machines,  the  number  of  turns 
per  coil  often  comes  out  a  very  small  number,  and  it  may  readily 
happen*  that  it  is  impossible  to  find  a  number  of  turns  with  either  star 
or  delta  connection  that  is  entirely  satisfactory.  In  this  case,  it  is 
frequently  desirable  to  arrange  the  winding  so  that  the  full  voltage  of  the 
machine  is  impressed  on  only  a  part  of  the  pole  windings.  The 
number  of  turns  can  then  frequently  be  adjusted  to  a  suitable  value. 
Thus  if  2.5  turns  were  required  with  a  star  winding,  the  corresponding 
number  for  a  delta  winding  would  be  4.33.  Neither  of  these  can  be 
attained.  By  connecting  only  half  the  poles  in  series,  the  number  of 
turns  per  coil  to  give  the  same  flux  density  is  increased  to  five  in  a  star 
winding,  and  the  difficulty  is  removed. 

In  a  case  like  that  mentioned  above,  where  the  number  of  turns 
per  coil  should  be  some  integer  plus  one-half,  as  6£,  9^,  etc.,  the 
difficulty  can  in  some  cases  be  overcome  by  winding  half  the  coils 
with  a  number  of  turns  corresponding  to  the  integer  of  the  number 
desired,  and  the  other  half  with  one  more  turn  each.  Thus  in  the 
case  referred  to,  half  of  the  coils  might  be  wound  with  two  turns,  and 
the  other  half  with  three  turns  each.  This  expedient  is  advisable 
only  in  case  the  number  of  slots  spanned  by  the  coils  is  odd.  If  this 
is  the  case,  it  will  be  found  that  when  the  top  of  a  slot  is  occupied  by 
a  coil  of  the  greater  number  of  turns,  the  bottom  coil  of  the  same  slot 
will  have  the  smaller  number.  Hence  the  number  of  conductors  in 
each  slot  will  be  the  same,  and  if  the  coils  are  inserted  in  such  a  way 
that  the  upper  coils  in  the  slots  have  alternately  the  larger  and  the 


154 


THE  INDUCTION  MOTOR 


smaller  number  of  turns,  the  windings  of  the  poles  will  be  found  to  be 
symmetrical,  and  the  winding  can  be  connected  up  in  any  of  the 
ordinary  ways. 

THE  FULL  OVERLAPPING  OR  BASKET  WINDING 

What  is  known  as  the  full  overlapping  or  basket  winding,  may  be 
regarded  as  a  special  case  of  the  above,  in  which  the  number  of  turns 
in  one  of  the  sets  of  coils  has  been  reduced  to  zero.  The  number 
of  coils  is  then  equal  to  half  the  number  of  slots,  and  the  same 
condition  as  given  above  holds  as  to  the  possible  pitch. 

The  advantage  of  this  winding  is  that  it  is  somewhat  easier  to  wind 
the  coils,  and  the  amount  of  insulating  material  is  less,  since  the  insula- 
tion between  coils  in  the  same  slot  is  done  away  with.  On  the  other 


FIG.  77.— Diamond  Shaped  Coils. 


FIG.  78. — "  Basket  "  or  Full  Overlapping 
Winding. 


hand,  somewhat  more  room  is  required  for  the  end  connections,  the 
inductance  of  the  end  connections  is  increased,  since  the  wires  are 
bunched  in  larger  groups,  and  the  improvement  in  the  dispersion 
coefficient  by  using  short-pitch  windings  cannot  be  realized  with 
this  type  of  coil.  The  saving  in  the  copper  of  the  end  connections 
by  the  use  of  the  fractional-pitch  winding,  is  of  course  retained.  This 
type  of  winding  is  rarely  used  except  in  the  case  of  small  machines. 
Fig.  77  shows  a  winding  employing  the  common  type  of  diamond- 
shaped  coils,  while  Fig.  78  shows  a  full  overlapping  coil. 

The  type  of  winding  shown  in  Fig.  79  is  known  as  a  concentric 
winding  or  spiral-coil  winding.  Its  great  disadvantage  is  that  it  requires 
two  or  more  forms  for  each  winding  instead  of  one.  Moreover,  it 
will  be  noted  that  some  of  the  coils  (in  a  three-phase  winding),  bend 
down  on  the  ends,  and  those  which  bend  down  on  one  end  are  straight 
on  the  other.  It  will  be  seen  at  once  that  the  coils  cannot  be  com- 
pletely formed  before  being  placed  in  the  slots,  but  that  the  coils  must 
in  general  be  cut  in  the  middle  of  one  end,  and  the  cut  ends  of  the 


FRACTIONAL-PITCH  WINDINGS 


155 


wires  united  one  by  one  after  the  coil  is  in  place.  This  results  in  the 
labor  cost  of  such  a  winding  being  very  high.  Their  principal  advan- 
tage is  that  the  ends  of  the  coils  may  readily  be  so  arranged  that  the 
different  coils  do  not  touch  at  any  point.  The  coils  may  then  be 


FIG.  79. — Concentric  Winding. 

easily  insulated  for  high  voltages,  and  the  only  point  liable  to  break 
down  is  within  the  slots.  This  type  of  winding  is  therefore  used 
principally  in  the  case  of  high-voltage  machines.  Some  of  these 
objections  do  not  apply  to  the  case  of  a  single-phase  winding,  and 
for  this  purpose  concentric  c6ils  are  commonly  used. 


CHAPTER  XI 

DESIGN  OF  A   50-H.P.,  750-REV.  PER  MIN.,  25-CYCLE  INDUCTION 
MOTOR 

IN  the  present  chapter  an  attempt  will  be  made  to  show  the  appli- 
cation of  some  of  the  principles  treated  in  the  preceding  chapters  to 
the  case  of  an  actual  motor.  For  this  purpose,  a  machine  of  moderate 
size  and  operating  at  a  favorable  speed  has  been  chosen.  We  should 
therefore  expect  to  be  able  to  design  a  motor  of  good  characteristics 
and  one  that  could  be  built  at  a  moderate  cost. 

In  designing  such  a  motor,  it  should  be  kept  in  mind  that  the 
same  frame  with,  if  possible,  the  same  slotting,  should  be  capable  of 
being  wound  for  either  two  or  three  phases,  and  for  the  ordinary 
commercial  voltages  of  220,  440,  and  550  volts.  A  machine  of  this 
size  would  rarely  be  wound  for  no  volts.  If  required  for  2200  volts, 
usually  an  entirely  different  design  using  larger  slots,  and  probably 
a  different  core  would  be  required.  It  is  perhaps  worthy  of  note  that 
two-phase  motors  are  by  no  means  so  frequently  required  as  are 
three-phase  ones,  and  this  is  particularly  true  in  the  case  of  25-cycle 
motors.  For  this  reason  we  will  work  out  the  design  for  three  phases. 

It  is  also  generally  advisable,  as  was  previously  pointed  out,  to 
build  at  least  two  ratings  of  motor  of  the  same  speed,  frequency,  etc., 
on  the  same  size  frame,  and  this  should  be  kept  in  mind  in  selecting 
the  appropriate  diameter. 

The  first  point  to  be  determined  is  the  output  coefficient,  as  upon 
the  value  of  this  depends  whether  the  design  shall  be  a  close  or  a  liberal 
one.  A  great  many  considerations  enter  into  the  determination  of  this 
constant,  such  as  the  general  policy  of  the  co  r.pany  manufacturing 
the  machine,  i.e.,  whether  it  is  desirable  to  build  a  motor  having  very 
liberal  characteristics  ad  giving  the  best  possible  service  to  the  cus- 
tomer, or  to  take  the  other  extreme,  one  that  can  be  built  at  the  lowest 
possible  cost  and  still  meet  the  heating  and  other  characteristics.  If 
the  machine  is  a  special  one,  and  only  one  of  the  rating  is  to  be  built, 
it  is  of  course  necessary  to  make  the  design  much  more  liberal  than 

156 


DESIGN  OF  A  CYCLE  INDUCTION  MOTOR  157 

would  otherwise  be  the  case,  since  if  the  design  should  be  unsuccess- 
ful, the  percentage  of  loss  would  be  much  greater  than  would  be  the 
case  if  a  great  number  were  to  be  manufactured  from  the  same  design. 
A  number  of  these  points  have  been  fully  treated  in  the  preceding 
pages. 

In  this  case,  the  motor  was  one  of  a  complete  line  of  75o-rev.  per 
min.,  25-cycle  motors,  and  from  the  results  of  the  motors  already 
tested  and  other  experience  it  appeared  that  an  output  coefficient 
of  about  38  was  the  best.  The  value  actually  used  employing  the 
nearest  -fa  in.  was  38.3.  From  the  formula 


38.3=* 
3    3 


we  find  at  once  that  the  value  of  d2l  is  1740.  Since  the  machine  has 
four  poles,  this  is  equivalent  to  saying  that  the  product  t2l=  1080, 
where  t  is  the  pole  pitch  and  /  the  active  length  of  iron,  not  counting 
the  air  ducts. 

Our  next  problem  is  to  determine  the  ratio  of  these  two  quantities. 
From  the  analysis  of  page  131,  it  will  be  seen  that  for  the  minimum 
amount  of  copper,  if  the  winding  is  to  be  full  pitch,  we  should  have 
t=%l.  This  would  give  the  dimensions  ^=9.9  and  £  =  3.3,  or  the 
length  of  active  iron  would  be  9.9  ins.,  and  the  diameter  of  the  rotor 
would  be  4.2  ins.  It  is  apparent  at  once  that  this  would  lead  to  a 
motor  having  very  poor  facilities  for  ventilation,  and  one  in  which 
the  power-factor  would  be  very  low.  In  fact  it  would  be  doubtless 
impossible  to  build  a  motor  of  these  relative  dimensions  and  using 
such  a  value  of  the  output  coefficient,  and  which  would  fall  within 
the  limit  of  40  degrees  C.  rise.  Without  further  consideration  we  can 
dismiss  the  idea  of  designing  the  motor  on  the  basis  of  minimum 
copper  cost. 

We  have  found  that  to  obtain  the  maximum  value  of  the  power- 
factor  we  should  design  the  motor  so  that  /=  (1.81  =0.9580)!.  In  this 
case  it  was  decided  to  use  slots  one-third  open  and  for  this  condition  we 
find  that  /=i-59/.  This  leads  to  the  values  £=7.55  and  ^=15.2. 
These  are  reasonable  values,  and  might  be  adopted.  In  this  case  on 
account  of  various  considerations,  such  as  the  size  of  frames  and 
punches  available,  and  the  fact  that  it  was  proposed  to  build  both  the  50- 
and  the  75-h.p.  motors  with  approximately  the  same  flux  densities  and 


158  THE  INDUCTION  MOTOR 

on  the  same  diameter  of  frame,  it  was  decided  to  use  a  somewhat  large 
diameter,  and  the  value  17  ins.  was  determined  upon.  This  gives  at 
once  6  ins.  for  the  value  of  the  net  core  length. 

As  was  previously  shown,  the  air  gap  should  be  as  short  as  is  con- 
sistent with  mechanical  considerations.  In  the  case  of  motors  having 
a  fairly  large  pole  pHch,  it  is,  however,  not  necessary  to  make  the  gap 
so  short,  in  order  to  obtain  reasonable  values  of  the  power-factor,  as 
would  be  the  case  with  motors  of  short  pole  pitch.  In  the  present  motor 
the  length  chosen  was  0.035  m-  A  shorter  gap  could  have  been  used 
if  necessary. 

The  number  of  ventilating  ducts  is  usually  so  taken  that  no  part 
of  the  active  iron  of  the  motor  is  more  than  about  one  inch  from  a 
radiating  surface.  This  leads  to  the  use  of  two  ducts  in  the  present 
case.  Each  duct  was  made  f  in.  wide.  In  a  larger  motor,  where  the 
air  would  be  forced  to  travel  a  greater  distance  past  the  laminations 
before  being  discharged  from  the  motor,  it  would  probably  be  better  to 
make  the  ducts  of  a  greater  width.  This  is  a  matter  to  be  settled  by 
experience. 

The  number  of  slots  chosen  for  the  stator  was  48,  or  4  slots  per  phase 
per  pole.  This  number  also  allows  of  winding  the  machine  with  the 
same  slotting,  as  a  two-phase  machine,  having  six  slots  per  phase  per 
pole.  Within  reasonable  limits,  the  greater  the  number  of  slots  the  bet- 
ter. Of  course  this  could  be  carried  too  far,  and  the  number  of  slots  in- 
creased to  so  great  a  number  that  the  percentage  of  insulating  material 
would  be  so  large  that  the  advantage  of  the  increased  number  of  slots 
would  be  more  than  counterbalanced.  It  must  also  be  kept  in  mind 
that  an  increase  in  the  number  of  slots  means,  in  general,  a  greater 
manufacturing  cost. 

Since  an  increase  in  the  number  of  slots  in  the  rotor  will  not  result 
in  so  great  an  increase  in  the  manufacturing  cost  as  would  the  same 
increase  in  the  stator,  it  is  customary  to  use  a  greater  number  of  rotor 
than  of  stator  slots.  It  is  also  customary  to  use  a  prime  number  of 
slots  in  the  rotor.  This  tends  to  prevent  magnetic  locking  of  the  stator 
and  rotor  with  consequently  little  or  no  starting  torque.  In  this  case, 
on  account  of  various  reasons,  no  was  chosen  as  the  number.  Previous 
experience  had  shown  that  although  not  a  prime  number,  it  would  give 
good  results.  The  dimensions  of  the  stator  and  the  rotor  slots  are 
shown  in  Fig.  3. 

The  factors  to  be  considered  in  determining  upon  the  proper  pitch 
of  the  coils  have  already  been  considered,  and  it  has  been  shown  (see 


DESIGN  OF  A  CYCLE  INDUCTION  MOTOR  159 

page   149)   that  the   most  favorable   pitch   is  given  by  the  formula, 

1.23$ 
l/P= — .     This  formula  is  represented  in  the   curve   of  Fig.  74. 

The  length  of  the  straight  part  of  the  coil  is  in  this  case  about  6|  ins. 
+ 1  \  ins.  =  8  ins.,  and  the  pole  pitch  is  13.4  ins.  It  is  therefore  seen  from 
the  curve  that  the  best  value  of  the  pitch  factor,  as  far  as  economy  of 
copper  is  concerned,  is  65  per  cent.  If  as  short  a  pitch  as  this  had  been 
chosen  it  would  have  caused  the  use  of  too  many  conductors  per  slot 
for  the*  size  of  slot  available,  and  on  this  account  a  somewhat  longer 
pitch  was  adopted.  The  pitch  actually  chosen  was  from  one  to  ten, 
or  75  per  cent. 

The  next  consideration  is  the  number  of  conductors  per  phase  to 
be  used  in  the  motor.  This  is  a  very  important  point  and  one  upon 
which  there  is  room  for  considerable  difference  of  opinion.  Other 
things  being  equal,  upon  the  number  of  conductors  depends  the  max- 
imum output  of  the  motor,  the  starting  torque,  and  the  point  at  which 
the  maximum  power-factor  will  occur.  The  principal  points  to  be  con- 
sidered have  already  been  treated.  American  practice  in  squirrel-cage 
machines  has,  to  a  certain  extent,  been  standardized,  and  in  the  case  of 
machines  in  which  the  power-factor  will  of  necessity  be  low  it  is  cus- 
tomary to  alow  a  maximum  output  of  at  least  200  per  cent  of  full  load. 
In  machines  of  large  pole  pitch  in  which  the  power-factor  will  be  high, 
it  is  in  the  author's  opinion  advisable  to  allow  a  greater  overload  capacity 
than  this. 

This  is  the  case  in  the  present  design,  and  a  maximum  output  of 
about  250  per  cent  is  desirable.  Before  the  number  of  turns  to  give 
this  can  be  computed,  it  is  necessary  to  estimate  the  dispersion  coefficient 
or  the  leakage  factor  in  any  of  the  ways  already  indicated. 

This  is  the  machine  already  referred  to,  and  its  leakage  coefficient 
and  dispersion  coefficient  have  already  been  computed  in  various  ways. 
Taking  as  the  value  of  "  C  "  the  number  8.95  (as  determined  on  page 
119),  using  this  number  and  substituting  in  the  formula  derived  on 
page  120,  we  find  as  the  maximum  value  of  the  k.w.  input  and  con- 
sequently the  approximate  h.p.  output, 

25,800X4X4402 

=  140   k.W. 


CbfN2(sm  r/2)2 

The  winding  used  is  to  be  connected  in  delta,  hence  the  value  440  is 
taken  as  for  voltage  per  phase.     Eleven  turns  per  coil  are  used,  or  the 


160  THE  INDUCTION  MOTOR 

number  of  conductors  per  phase  is  352.  As  was  previously  explained, 
the  winding  would  usually  be  star  connected.  In  this  case  the  delta 
winding  seems  to  be  preferable  on  account  of  giving  a  more  suitable 
value  to  the  maximum  power  input.  This  matter  was  fully  discussed 
on  page  152.  The  factor  0.925  or  sin  (i|-^)°=  0.925,  is  used  on  account 
of  the  short-pitch  coils  (see  page  147). 

At  full  load  and  average  efficiency  and  power-factor  the  motor  will 
take  approximately  62  amperes.  The  current  per  conductor  in  a  delta- 
connected  winding  will  be  the  line  current  divided  by  V^or  in  this  case 
35.8  amperes.  It  will  be  found  that  22  number  6  wires  can  be  readily 
accommodated  in  the  slot  chosen.  Using  this  size  of  wire  we  have 
26,250-=-  35.8  =  735  circular  mils  per  ampere.  In  many  designs  this 
value  is  taken  as  low  as  600.  In  this  case,  since  the  rotor  loss  is 
rather  large  the  stator  copper  loss  was  kept  low. 

The  maximum  value  of  the  flux  density  in  the  gap  is  found  from 
the  equation, 


Jj    =  -  =  —  ---  '  =  49  700 

'    ANfsinr/2     6X13.4X25X352X0.925 

The  total  flux  per  pole  is  given  by 


—  =  2.54X106. 


The  cross-section  back  or  the  slots,  making  10  per  cent  allowance  for 
the  waste  space  between  the  laminations,  is  readily  found  to  be  15.5 
sq.in.,  and  from  this  the  maximum  flux  density  in  the  iron  is  81,600. 
This  would  be  excessively  high  in  the  case  of  a  6o-cycle  motor,  but  is 
satisfactory  in  the  case  of  a  25-cycle  machine.  This  is  so  since  for  the 
same  flux  density  the  hysteresis  loss  will  be  reduced  in  the  ratio  of  25 
to  60,  and  the  eddy-current  loss  in  the  proportion  of  the  square  of  this 
ratio.  It  is  decidedly  advantageous  to  have  the  density  in  the  iron  as 
high  as  possible,  since  by  cutting  down  the  cross-section  of  the  iron,  we 
reduce  greatly  the  outside  diameter  of  the  motor,  and  consequently  its 
weight  and  cost. 

The  flux  densities  in  the  teeth  of  the  stator  and  the  rotor  are  readily 
obtained  by  multiplying  the  maximum  value  of  the  flux  density  in  the 
air  gap,  by  the  ratio  of  the  slot  pitch  to  the  tooth  section.  The  tooth 
pitch  is,  of  course,  to  be  taken  at  the  air  gap,  and  the  tooth  width  at 
the  point  for  which  we  wish  to  calculate  the  density.  Some  designers 
would  increase  the  value  thus  obtained  about  10  per  cent  to  allow  for 


DESIGN  OF  A  CYCLE  INDUCTION  MOTOR  161 

the  spaces  between  the  laminations.  In  this  case,  the  density  is  found  to 
be  122,000  lines  per  square  inch  at  the  root  of  the  tooth,  and  139,000 
at  the  narrowest  point.  These  values  are  about  as  high  as  is  allowable 
both  on  account  of  the  large  losses  at  high  densities,  and  to  even  a 
greater  extent  on  account  of  the  large  magnetizing  current  required  to 
force  the  flux  through  the  teeth.  This  consideration  is  less  important 
in  the  case  of  machines  in  which  the  power-factor  is  naturally  high, 
i.e.,  in  the  case  of  machines  of  large  pole  pitch. 

The  iron  loss  is  determined  in  the  manner  described  on  page  137. 
The  weight  of  the  stator  iron  is  computed,  no  allowance  being  made  for 
the  teeth,  and  the  average  loss  per  pound  for  the  given  flux  density 
determined  from  the  curve.  In  this  case  the  loss  was  taken  as  3.8 
watts  per  pound.  The  test  of  the  machine  after  it  had  been  built 
showed  a  total  loss  of  1325  watts,  or  2.79  watts  per  pound.  It  might 
be  again  pointed  out,  that  this  is  far  greater  than  the  loss  which  would 
be  expected  from  a  test  of  the  iron  with  uniform  flux  distribution.  For 
example,  this  particular  iron  showed  under  test  a  loss  of  less  than  one 
watt  per  pound,  at  the  same  flux  density  as  that  employed  in  the  motor. 

The  determination  of  the  current  per  rotor  bar  is  perhaps  most 
readily  made  by  a  consideration  of  the  total  sheet  of  current  in  the 
stator  and  that  in  the  rotor.  If  the  efficiency  and  power-factor  were 
the  same  in  both,  these  sheets  would  be  the  same,  that  is  the  arithmetical 
sum  of  all  the  currents  in  the  stator  and  of  all  those  in  the  rotor  would 
be  the  same.  While  this  statement  would  require  some  modification 
to  be  strictly  true,  it  is  nearly  enough  correct  for  our  purpose. 

To  make  use  of  this  fact,  we  assume  approximately  the  product 
of  the  efficiency  and  power-factor  in  the  rotor,  and  calculate  the  current 
per  stator  conductor  on  the  assumption  that  the  power-factor  and 
efficiency  in  the  stator  are  the  same  as  the  rotor.  The  product  of 
this  current  into  the  total  number  of  stator  conductors,  gives  the  stator 
current  sheet.  This  value  divided  by  the  number  of  rotor  bars  gives 
the  rotor  current  per  bar.  In  this  case,  we  may  take  po we r-f actor  X 
efficiency  =  0.92.  The  current  per  stator  conductor  would  then  be 
30.7  and  current  per  rotor  bar =30.7X352X3  -=-110  =  295.  The  size 
of  the  rotor  bars  was  taken  as  0.1574  in.  by  0.787  in.  This  gives  a 
cross-section  of  513  circular  mils  per  ampere. 

The  current  per  end  ring  is  given  by  the  formula 

295X110X2 
— 
SXn 


162  THE  INDUCTION  MOTOR 

The  sheet  of  current  under  each  pole  divides  and  half  passes  in  each 

1 10 

direction.     The  number  of  bars  under  half  a  pole  is  in  this  case . 

2X4 
If  the  virtual  current  per  bar  is  /  the  maximum  current  (assuming 

2  v/2 

sinusoidal  distribution)  is  A/2/  and  the  average  current  will  be 1. 

If  the  number  of  bars  per  half  pole  be  n,  the  maximum  current  in  the 

end  ring  will  be —nl  and  the  virtual  value  will  be  — nl. 

n  n 

The  size  of  the  rings  was  taken  as  f  in.  by  ij  ins.  The  material 
was  cast  copper.  The  cross-section  is  539  circular  mils  per  ampere. 
This  is  liberal,  and  the  density  might  have  been  made  somewhat 
higher  without  danger  of  heating  in  normal  operation. 

The  active  length  of  the  rotor  bars,  i.e.,  the  part  carrying  current 
is  g\  ins.,  and  the  weight  of  the  active  rotor  copper  is  readily  found  to 
be  41.2  Ibs.  The  outside  diameter  of  the  ring  was  16  ins.  and  the  inside 
diameter  13.5  ins.  The  weight  of  both  rings  is  then  31.6  Ibs. 

The  losses  in  the  bars  at  full  load  are  given  by 

(700\ 2  /700\ 2 

-)  =4I.2X8X(  — )  =5^8  watts. 
Aa)  \533/ 

The  loss  in  the  end  rings  is  likewise  given  by  the  formula: 

3i.6X8X3-65x(— )  =1550  watts. 
\539/ 

In  this  the  factor  3.65  has  been  introduced,  since  the  resistance  of  the 
cast  copper  used  was  approximately  3.65  times  as  great  as  is  that  of 
drawn  copper  as  used  in  wires  and  rotor  bars.  This  ratio  of  course 
varies  somewhat  in  different  samples,  depending  upon  the  amount  of 
impurities,  methods  of  casting,  etc.  The  total  loss  in  the  rotor  at  full 
load  is  the  sum  of  these  two,  or  2118  watts. 

The  magnetizing  current  is  readily  calculated  from  the  formula 
given,  but  before  doing  so  it  is  necessary  to  estimate  what  we  may  call 
the  equivalent  length  of  the  air  gap.  This  is  made  necessary  on  account 
of  the  fact  that  the  flux,  instead  of  being  distributed  in  a  uniform  man- 
mer  in  the  air  gap  is,  on  account  of  the  teeth,  distributed  in  a  series  of 
tufts.  The  density  in  these  tufts  is,  of  course,  higher  than  would  be 
the  case  if  there  were  no  projections,  and  consequently  a  higher  m.m.f. 


DESIGN  OF  A  CYCLE  INDUCTION  MOTOR  163 

is  necessary  to  force  the  flux  across  the  gap.  This  is  equivalent  to 
having  a  longer  air  gap. 

Methods  of  estimating  the  factor  by  which  the  length  of  the  gap 
must  be  multiplied  to  correct  for  this  tufting  may  be  found  in  many  of 
the  text-books  devoted  to  designing.  In  general  we  may  say  that  there 
is  no  absolutely  correct  method  of  estimating  this  factor.  As  a  matter 
of  fact,  in  the  case  of  the  induction  motor,  it  is  not  essential  that  we 
have  such  a  method,  since  on  account  of  the  shortness  of  the  gap, 
variations  in  the  length  of  the  gap  are  sure«  to  occur  and  change  so 
greatly  the  magnetizing  current  that  an  exact  method  would  lie  super- 
fluous. Perhaps  as  simple  a  method  as  any  is  to  assume  that  all  the 
lines  of  induction  on  leaving  the  stator  teeth  pass  in  a  straight  line 
directly  across  to  the  rotor.  This  assumption  will  be  only  approximately 
true  even  in  the  case  of  an  induction  motor  where  the  air  gap  is  very 
short,  and  would  be  far  indeed  from  the  truth  in  the  case  of  an  ordinary 
direct-current  machine. 

In  the  present  case,  the  slot  pitch  in  the  stator  is  1.113  m->  and 
since  the  slot  opening  is  0.25  in.,  the  exposed  iron  per  slot  is  0.863  in. 
The  ratio  of  the  slot  pitch  to  the  exposed  iron  per  slot  is  therefore  1.29. 
In  the  same  way  the  factor  for  the  rotor  is  found  to  be  i.n,  or  the  factor 
by  which  we  must  multiply  is  the  product  of  these  two  or  1.43.  Using 
this  for  the  no-load  current,  we  get 


.  0.243  X49> 

Nismn/2  352X0.925 

The  actual  test  of  the  machine  after  construction  gave  a  no-load  current 
of  13.9  amperes.  The  magnetizing  current  or  the  wattless  component 
of  this  was  13.6  amperes.  This  agreement  is  as  good  or  better  than 
can  be  expected  in  the  regular  course  of  manufacturing,  especially  since 
no  allowance  was  made  for  the  ampere-turns  required  to  drive  the 
flux  through  the  teeth  and  other  iron  of  the  machine. 

Using  the  factor  we  have  just  determined  for  correcting  the  length 
of  the  air  gap,  we  find  the  equivalent  length  to  be  0.035  in-  X  1.43  =0.05 
in.  Using  this,  the  value  of  a,  the  leakage  coefficient,  is  readily  found 

d  0.05 

from  the  equation,  a  =  C—  =  8.95X  -  =0.0333.     From  this  the  max- 
t  13.4 

imum  value  of  the  power-factor  is  given  by 


164  THE  INDUCTION  MOTOR 

Having  the  losses  in  the  rotor  at  full  load,  we  can  readily  calculate 
the  starting  torque  of  the  motor.  We  have  found  that  the  maximum 
power  the  motor  is  capable  of  taking  is  140  k.w.,  and  this  corresponds 
to  a  starting  current  of  367  amperes.  The  ratio  of  the  starting  current 
to  the  full-load  current  is  367-7-62  =  5.92.  The  rotor  loss  will  be 
approximately  proportional  to  the  square  of  the  primary  current,  and 
consequently  the  loss  at  start  will  be  21 18X5.922=  75,000  watts. 

This  is  the  starting  torque  in  synchronous  watts.  The  torque 
in  synchronous  horse-power  is  this  number  divided  by  746  or  101  h.p. 
We  most  frequently,  however,  desire  the  starting  torque  in  percentage 
of  the  normal  full-load  torque.  Since  the  actual  full-load  torque  would 
be  different  for  each  motor  of  the  same  rating  depending  upon  the 
slip  at  full  load,  it  is  customary  to  calculate  the  starting  torque  in  per- 
centage of  the  torque  the  motor  wouid  have  to  develop  to  deliver  full 
load  if  it  were  to  operate  at  synchronism.  In  the  present  case,  this  is 
found  by  dividing  the  starting  torque  in  synchronous  horse-power  by 
the  horse-power  of  the  motor  or  202  per  cent. 

The  slip  is  readily  obtained  by  dividing  the  loss  in  the  rotor  at  full 
load  by  the  output  of  the  rotor,  or  5.67  per  cent. 

The  stator  copper  loss  is  readily  obtained  from  the  weight  of  wire 
and  the  circular  mils  per  ampere,  and  is  found  to  be  1116  watts.  The 
friction  loss  can  only  be  estimated  from  other  machines  built  on  the 
same  frame  and  operating  at  the  same  speed.  In  this  case  we  may 
assume  it  as  being  500  watts. 

We  now  have  all  the  losses,  and  adding  these  to  the  output  we 
obtain  the  input.  The  output  divided  by  the  input  gives  the  efficiency. 
In  this  case  it  is  87  per  cent. 

The  comparison  of  the  fixed  and  variable  losses  is  of  great  interest, 
since  the  maximum  efficiency  will  be  obtained  when  these  two  are 
equal.  The  fixed  losses  consist  of  the  iron  loss  plus  the  friction  loss, 
and  amounts  to  11,800  plus  500  or  2300  watts.  The  variable  losses 
are  the  copper  losses  in  the  stator  and  rotor,  or  1116  plus  2118,  or  3234. 
The  best  efficiency,  as  can  be  readily  shown,  will  be  obtained  at  a  load 
of  42.5  h.p.,  and  at  this  load  it  will  be  87.4  per  cent. 

The  best  point  at  which  to  have  the  efficiency  occur,  is  influenced 
by  a  variety  of  factors.  If  the  load  is  frequently  a  light  one,  it  is  of 
course  advisable  to  have  the  point  of  best  efficiency  occur  somewhat 
before  full  load.  If  on  the  other  hand  the  motor  is  so  designed  that 
the  efficiency  at  full  load  and  at  125  per  cent  of  full  load  are  equal,  the 
best  efficiency  occurring  at  some  intermediate  point,  the  over-load 


DESIGN  OF  A  CYCLE  INDUCTION  MOTOR  165 

capacity  of  the  machine  will  be  good.  The  truth  of  this  will  appear 
if  we  consider  that  since  the  efficiency  is  the  same  as  full  load  and  at 
125  per  cent  of  full  load  the  losses  at  the  two  points  will  be  in  the  pro- 
portion of  four  to  five.  Hence  if  the  rise  in  temperature  under  full 
load  is  not  more  than  40  degrees  centigrade,  the  rise  under  125  per  cent 
load  will  not  be  more  than  50  degrees  centigrade.  While  it  would 
probably  not  be  advisable  to  operate  constantly  at  this  rise  in  tempera- 
ture, yet  its  application  for  a  reasonable  length  of  time  will  do  little 
or  no  harm.  From  the  standpoint  of  the  manufacturer,  this  property 
allows  him  to  guarantee  excellent  overload  characteristics. 

The  standard  overload  guarantee  is  a  rise  of  55  degrees  centigrade 
for  a  run  of  two  hours  at  125  per  cent  of  full  load,  the  overload  run  to  be 
made  immediately  following  a  full  load  run  of  sufficient  duration  so 
that  the  motor  has  attained  its  final  temperature.  Hence  a  motor  so 
designed  that  the  efficiency  is  the  same  at  full  load  and  125  per  cent 
of  full  load,  will  do  far  better  than  this  guarantee.  On  the  other  hand, 
in  the  case  of  a  motor  in  which  the  best  efficiency  is  attained  at  about 
75  per  cent  of  full  load,  the  losses  will  run  up  to  a  far  greater  value  on 
over  load,  and  the  rise  in  temperature  will  approximate  that  of  the 
standard  guarantee. 

The  rise  in  temperature  of  the  motor  must  next  be  estimated,  and 
this  can  be  done  in  any  one  of  the  ways  described.  For  example,  in  the 
case  of  the  frame  used,  it  was  known  from  previous  experience  that 
at  the  peripheral  speed  employed  (3340  feet  per  minute),  approximately 
4250  watts  could  IDC  radiated  without  causing  a  rise  of  more  than  40 
degrees  centigrade.  Since  it  was  the  intention  to  use  small  wings  on 
the  rotor,  about  25  per  cent  more  than  this  could  be  radiated.  The  com- 
puted loss,  less  the  friction  of  the  bearings  and  the  air  friction,  is  5034 
watts.  Hence  the  machine  should  be  safe  as  far  as  heating  is  concerned. 

As  a  check  on  the  above  the  radiating  surface  of  the  core  was  com- 
puted counting  one  side  of  the  air  gap  and  one  side  of  each  of  the  ven- 
tilating ducts,  thus, 

7T 

(25.52— io2) — h7r6X(io+i7  +  25.5)  =  273o  sq.in. 

This  gives  1.84  watts  per  square  inch,  and  from  previous  experience 
it  was  known  that  approximately  1.5  watts  per  square  inch  could  be 
radiated  without  wings  or  perhaps  2.25  with  the  small  wings  employed. 
The  actual  rise  under  load  was  29.3  degrees  in  the  iron  and  38.3  degrees 


166  THE  INDUCTION  MOTOR 

in  the  copper.  The  actual  losses  less  friction  were  5165  watts.  This 
•would  indicate  that  the  frame  would  radiate  6250  watts  with  a  rise 
of  40  degrees.  Dividing  this  by  the  radiating  surface  2730  sq.in.  gives 
the  value  of  2.29  watts  per  square  inch.  These  values  of  watts  for  the 
frame  and  watts  per  square  inch  may  now  be  plotted  by  the  designer 
on  his  private  curves,  for  future  reference  and  as  a  check  on  his  previous 
values.  In  this  way,  a  mass  of  valuable  information  is  gradually 
built  up. 


CHAPTER  XII 

SPECIAL  TYPES  OF  MOTORS 

IN  the  following  some  of  the  special  types  of  motors  which  have 
been  developed  to  meet  certain  needs  will  be  discussed.  In  order  to 
keep  the  treatment  brief,  the  motors  mentioned  will  be  only  such 
as  are  manufactured  in  the  United  States. 

Most  of  the  motors  which  are  here  designated  as  special  types, 
have  been  developed  in  the  effort  to  combine  the  advantages  of  the 
squirrel-cage  and  the  wound-rotor  types.  Where  the  service  does  not 
present  difficulties  in  regard  to  starting  torque,  and  variable  speed  is  not 
required,  the  former  is  in  every  respect  the  better.  Its  cost  is  less, 
it  is  of  more  simple  and  robust  construction,  its  power-factor  will  average 
from  one  to  two  per  cent  better.  If  constructed  with  no  regard  to 
starting  torque,  the  efficiency  will  also  be  better.  If  from  150  to  200 
per  cent  starting  torque  must  be  provided,  a  high  resistance  rotor  will 
be  required,  and  this  will  probably  reduce  the  efficiency  to  no  more  or 
slightly  less  than  that  of  a  wound-rotor  machine. 

If  on  the  other  hand  the  proposed  application  is  one  which  requires 
frequent  starting  under  heavy  loads,  or  if  it  is  necessary  to  limit  the 
demand  for  power  from  the  line  or  if  the  speed  of  the  motor  must  be 
readily  capable  of  being  varied  through  a  large  range,  the  wound  rotor 
type  is  the  one  to  be  used. 

The  General  Electric  type  L  induction  motor  is  one  of  the  many 
attempts  to  combine  the  advantages  of  the  wound-rotor  and  the  squirrel- 
cage  types.  The  stator  is  of  the  standard  type,  differing  in  no  way  from 
one  that  would  be  used  with  a  squirrel-cage  motor.  The  rotor,  how- 
ever, is  provided  with  a  three-phase  winding  and  a  three-phase  starting 
resistor  is  mounted  inside  the  rotor  itself.  Three  brushes  are  arranged 
in  such  a  manner  that  they  make  contact  directly  with  the  resistors. 
The  brushes  are  movable  from  the  outside  by  means  of  a  rod  passing 
through  the  hollow  shaft.  By  pressing  this  into  the  rotor,  the  successive 
sections  of  the  resistor  are  short-circuited,  and  the  motor  is  thus 
gradually  brought  up  to  speed. 

167 


168  THE  INDUCTION  MOTOR 

This  type  of  motor  is  entirely  self  contained,  and  no  external  appli- 
ances are  required  with  the  exception  of  a  line  switch  and  the  line 
fuses.  The  absence  of  an  external  starter  compensates  at  least  in  part 
for  the  increased  cost  of  the  rotor  construction  over  that  of  a  squirrel- 
cage  rotor.  In  fact,  at  the  present  time,  the  General  Electric  Company 
offer  the  two  types  for  approximately  the  same  price.  This  price  is 
some  25  or  30  per  cent  lower  than  the  cost  of  a  corresponding  wound- 
rotor  machine  with  its  starting  resistor,  and  hence,  in  applications  where 
it  can  be  used,  it  has  a  great  advantage  over  either  of  the  usual  types. 

As  an  offset  to  these  advantages,  we  have  the  fact  that  the  rotor 
is  of  much  frailer  construction,  and  is  consequently  more  liable  to 
mechanical  injury  or  electrical  breakdown.  As  was  pointed  out,  the 
power-factor  is  lower  than  that  of  the  corresponding  squirrel-cage 
motor,  and  the  pull  out  point,  and  hence  the  overload  capacity,  is  less. 
We  have  also  the  disadvantage  that  the  radiation  of  heat  from  the  rotor 
is  somewhat  impeded  by  the  resistors  mounted  there.  The  efficiency 
of  the  two  types  is  about  the  same,  the  internal  starter  motor  having 
perhaps  some  slight  advantage. 

This  form  of  motor  can  not  be  used  for  work  where  variable  speed 
is  required.  On  account  of  the  fact  that  the  resistors  are  mounted  in 
the  rotor,  it  would  be  impossible  to  get  rid  of  the  heat  generated  if  the 
motor  were  to  be  used  under  the  condition  of  full  torque  with  all  or 
part  of  the  resistors  in  circuit. 

THE  WAGNER  TYPE  BW  INDUCTION  MOTOR 

The  Wagner  type  BW  induction  motor  combines  the  advantages 
of  the  wound  rotor  and  squirrel-cage  types  of  motor.  That  is,  it  has 
approximately  the  starting  and  running  characteristics  of  the  G.  E. 
type  L  motor  but  performs  its  functions  automatically.  Fig.  81  is  a 
general  view  of  the  motor,  and  Fig.  82  shows  somewhat  the  construction 
of  the  rotor. 

The  stator  presents  no  particular  features,  being  of  the  same  con- 
struction as  would  be  used  in  a  squirrel-cage  machine.  The  rotor, 
however,  has  a  winding  resembling  that  of  a  direct-current  machine, 
the  ends  of  the  coils  being  carried  out  to  the  bars  of  a  commutator. 
The  winding  is  so  connected  that  during  the  starting  period  a  portion 
of  the  winding  opposes  its  e.m.f.  to  that  of  the  remainder.  Suppose, 
for  example,  that  the  coils  are  so  connected  that  one-third  of  the  coils 
oppose  the  action  of  the  remaining  third.  The  result  will  be  the  same 


SPECIAL  TYPE  OF  MOTORS 


169 


as  though  the  motor  were  wound  with  one-third  of  the  turns  actually 
employed,  and  a  resistance  equal  to  the  resistance  of  the  remaining 
two-thirds  of  the  winding  were  connected  in  series  with  it. 

As  soon  as  the  motor  attains  a  certain  predetermined  speed,  a 
centrifugal  weight  acts  to  press  a  short-circuiting  device  against  the 
commutator.  In  this  way,  each  of  the  coils  is  short-circuited  and  the 
motor  acts  in  much  the  same  manner  as  a  squirrel-cage  machine. 

Although  a  commutator  was  mentioned  as  being  used  in  connection 
with  the  short-circuiting  of  the  machine,  it  should  not  be  understood 


FIG.  81. — Wagner  Type  of  BW  Induction  Motor. 

that  the  commutator  is  used  in  the  ordinary  way.  It  serves  merely  as  a 
convenient  means  of  connecting  the  coils  in  the  two  ways  mentioned. 
No  brushes  are  employed  and  the  device  is  in  no  sense  used  as  a  com- 
mutator. The  construction  of  the  centrifugal  device  is  the  same  as 
that  employed  on  the  well-known  single-phase  motors  of  this  company's 
manufacture,  except  that  the  brushes  and  brush  mechanism  are  omitted. 
The  short-circuiting  device  is  the  same. 

The  only  accessories  required  with  this  type  of  motor  are  a  three  - 
or  four-pole  main  line  switch  and  the  corresponding  fuses.  The  action 
of  the  motor  is  entirely  automatic.  To  start,  all  that  is  necessary  is 
to  close  the  main  line  switch.  At  the  proper  speed,  the  centrifugal 


170  THE  INDUCTION  MOTOR 

device  will  operate  and  short-circuit  the  rotor.  To  stop,  the  main  line 
switch  is  opened.  The  motor  will  therefore  take  care  of  itself  in  case 
the  line  voltage  is  interrupted  from  some  cause,  and  later  restored  without 
warning.  The  motor  is  well  suited  for  distant  control  as  the  main  line 
switch  may  be  located  as  any  convenient  point  and  the  motor  started 
and  stopped  from  that  point.  It  is  likewise  well  adapted  to  the  opera- 
tion of  automatic  pumping  systems  where  the  circuit  is  opened  and  closed 
by  the  movements  of  a  float  or  by  similar  means. 

The  starting  characteristics  of  a  35  h.p.,  1800  rev.  per  min.,  three- 
phase  6o-cycle  motor  of  this  type  as  furnished  by  the  manufacturer  are 
shown  in  Fig.  83. 


FIG.  82.— Rotor  of  Wagner  Type  BW  Induction  Motor. 

Let  us  assume  that  one  of  these  motors  has  such  a  rotor  resistance 
that  the  slip  at  full  load  would  be  3^  per  cent.  At  standstill  if  it  were 
of  the  usual  construction  and  full-load  current  were  passed  through  it 
the  rotor  loss  would  likewise  be  approximately  3^  per  cent.  This  is 
on  the  assumption  that  the  ratio  of  the  stator  and  rotor  currents  is  the 
same  at  standstill  as  at  full  load,  which  is  nearly  the  case.  Under  these 
circumstances  the  starting  torque  would  be  the  same  as  the  rotor  loss 
or  3^  per  cent.  In  the  case  of  this  motor,  however,  since  one-third  of 
the  winding  is  opposed  to  the  other  two-thirds,  we  may  consider  that 
it  is  equivalent  to  a  motor  with  one-third  as  many  rotor  turns,  or  with  a 
given  stator  current  the  rotor  current  will  be  three  times  as  large.  Since 
the  loss  is  proportional  to  the  square  of  the  rotor  current,  the  loss  will  be 
nine  times  as  large  or  the  starting  torque  will  be  9X3^=31^  per  cent 


SPECIAL  TYPE  OF  MOTORS 


171 


with  full-load  current.  If,  however,  the  motor  takes  say  three  times 
full-load  current  when  thrown  on  the  line,  as  is  the  case  with  the  motor, 
the  characteristics  of  which  are  shown  here,  the  starting  torque  will  be 
increased  in  the  proportion  of  the  square  of  three  or  it  will  be  283  per 
cent.  This  is  approximately  the  same  as  that  of  the  motor  shown. 

On  a  curve  of  speeds  and  torque,  not  reproduced  here,  the  actual 
slip  of  the  motor  is  given  as  9  per  cent  instead  of  3^  per  cent.  This 
discrepancy  is  probably  due  to  two  causes.  In  the  first  place  the  short- 
circuiting  device  undoubtedly  introduces  some  resistance  into  the  rotor 
circuit  and  thus  causes  the  slip  to  be  somewhat  greater  than  would 


100 


al  Starting  Characteristics 
o^VVagner  D  W  Motor 


30 


40  60  80 

Per  Cent  Synchronous  Speed 


FIG.  83.  —  Starting  Current  of  Wagner  BW  Motor. 


otherwise  be  the  case.  The  writer  has  no  means  of  estimating  the 
importance  of  this  effect,  but  doubtless  the  increase  of  resistance  is 
small. 

In  addition  to  this  cause  of  increased  slip  we  have  the  fact  that  during 
starting  we  have  a  more  or  less  disturbed  condition  of  the  flux.  This 
is  due  to  the  fact  that  the  turns  through  which  the  current  is  in  the  wrong 
direction  do  not  exactly  offset  the  inductance  of  the  third  of  the  winding 
to  which  they  are  opposed.  The  effect  is  much  the  same  as  though 
the  ratio  of  turns  when  short-circuited  to  the  turns  when  not  short- 
circuited  were  less  than  i  to  3.  Thus  the  rotor  current  and  the  starting 
torque  are  somewhat  reduced. 


172  THE  INDUCTION  MOTOR 

As  is  apparent  from  the  diagram,  the  current  decreases  as  the  motor 
approaches  synchronism,  and  if  the  resistance  remained  the  same  a 
condition  would  soon  be  reached  where  the  torque  would  be  so  much 
reduced  that  the  motor  would  cease  to  accelerate  further.  The  centrifu- 
gal device  should  be  so  set  that  before  this  point  is  reached  the  rotor 
winding  is  shbrt-circuited.  This  short-circuiting  should,  however,  be 
done  as  late  as  possible  as  the  rush  of  current  when  the  centrifugal 
device  operates  will  then  be  less. 

The  principal  merits  of  this  device  have  already  been  mentioned. 
Its  principal  defects  as  compared  with  a  squirrel-cage  motor  are  that  the 
efficiency  and  the  power-factor  are  both  appreciably  lower.  In  addi- 
tion the  motor  is  by  no  means  so  simple,  and  is  of  course  more  liable 
to  accident.  That  the  power-factor  is  lower  is  apparent  from  the  fact 
that  the  winding,  although  it  is  short-circuited,  still  presents  many 
factors  in  common  with  the  wound-rotor  type  of  machine.  The  latter, 
as  we  have  seen,  has  a  leakage  coefficient  about  33  per  cent  greater  than 
a  corresponding  squirrel-cage  motor.  This  type  of  machine  is  inter- 
mediate in  its  characteristics,  and  its  coefficient  may  perhaps  be  15  per 
cent  greater.  Thus  in  the  example  given,  the  power-factor  is  92.5 
per  cent  at  its  maximum  value.  With  an  improvement  of  15  per  cent 
in  the  leakage  coefficient,  this  value  would  be  raised  to  about  93.3 
per  cent.  The  difference  is  not  striking  in  this  case  since  a  four-pole 
motor  of  this  rating  is  inherently  of  high  power-factor. 

That  the  efficiency  is  low  follows  from  the  fact  that  it  is  necessary 
to  allow  a  considerable  slip  in  order  that  the  starting  torque  may  be 
great  enough.  Thus  the  full-load  efficiency  is  given  as  83  per  cent. 
The  full-load  loss  is  consequently  1 7  per  cent.  Of  this  the  rotor  copper 
loss  accounts  for  9  per  cent,  leaving  only  8  per  cent  for  the  stator  copper 
loss  and  the  iron  loss. 

In  comparison  with  the  wound-rotor  type  of  machine,  this  machine 
is  somewhat  inferior  in  that  the  starting  torque  is  not  adjustable.  That 
is,  the  motor  must  take  its  maximum  current  at  the  start  even  though 
the  load  be  very  light.  This  is,  of  course,  not  a  very  serious  objection. 
One  of  more  importance  is  the  fact  that  the  torque  drops  off  materially 
as  the  speed  increases.  If  the  starting  torque  required  were  say  150 
per  cent  and  this  torque  had  to  be  maintained  up  to  full  speed,  it  would 
be  necessary  to  set  the  centrifugal  device  so  that  the  winding  would  be 
short-circuited  at  65  per  cent  of  full  speed  instead  of  at  about  85  per 
cent.  This  would  of  course  give  rise  to  a  large  rush  of  current  at  the 
moment  when  the  short-circuiting  device  operated.  Moreover,  a  load 


SPECIAL  TYPE  OF  MOTORS 


173 


requiring  nearly  the  maximum  starting  torque  of  the  motor  at  all  speeds 
could  not  be  started  at  all  by  this  type  of  motor.  The  conditions  men- 
tioned are  not  of  frequent  occurrence,  but  if  present  would  require  the 
use  of  a  wound-rotor  machine. 

THE  BURKE  INDUCTION  MOTOR 

The  Burke  Electric  Company  manufacture  an  interesting  type  of 
squirrel-cage  induction  motor  in  which  the  changes  from  the  usual 
type  are  made  apparently  not  for  the  sake  of  obtaining  improved  opera- 
ting characteristics  but  for  the  sake  of  an  improved  construction  from  a 


FIG.  84. — Rotor  of  Burke  Induction  Motor. 

mechanical  standpoint.  Like  most  of  the  other  motors  here  described, 
the  changes  are  in  the  rotor.  A  general  view  of  the  rotor  is  shown  in 
Fig.  84,  and  the  construction  of  the  rotor  bars  will  be  apparent  from 
Fig.  85.  Instead  of  using  the  usual  bar  winding,  the  rotor  conductors 
are  made  by  cutting  a  slit  in  a  wide  bar  of  copper,  and  pulling  this  out 
into  the  familiar  form  of  a  diamond  shaped  stator  coil.  These  are  then 
inserted  in  the  slots  in  much  the  same  way  as  is  done  in  the  case  of  a 
stator  winding,  and  are  secured  by  wedges. 

The  advantage  of  this  type  of  winding  is  that  since  neither  soldered 
or  bolted  joints  are  used,  it  is  practically  impossible  to  burn  out  the 
rotor  winding.  On  the  other  hand,  there  is  the  disadvantage  that  on 
account  of  the  greater  length  of  the  end  connections,  the  leakage  coef- 
ficient of  the  machine  is  greater  than  would  be  the  case  with  a  squirrel- 


174  THE  INDUCTION  MOTOR 

cage  winding.  As  has  been  explained,  this  results  in  lower  power- 
factor,  slightly  lessened  efficiency  and  a  considerably  lower  starting 
torque  and  pull  out  point.  Using  bars  of  pure  copper  in  this  way,  it 
would  also  appear  difficult  in  many  cases  to  obtain  sufficient  resistance 
in  the  rotor  to  give  the  desired  starting  torque.  Of  course  this  could 
be  obtained  by  reducing  the  section  of  the  bars  to  the  proper  point, 
but  this  has  the  disadvantage  that  the  heat  developed  on  account  of  the 


FIG.  85. — Rotor  Coil  of  Burke  Induction  Motor. 

rotor  loss  is  largely  concentrated  inside  the  slots.  It  is  of  course  pref- 
erable to  have  this  heat  generated  in  the  end  rings,  as  it  is  much 
more  easily  radiated  from  the  rings  than  would  be  the  care  if  it  were  com- 
pelled to  travel  some  distance  before  coming  to  a  surface  exposed  to  the 
air. 

SQUIRREL-CAGE  CRANE  MOTORS 

Motors  of  this  type  were  formerly  used  to  a  considerable  extent 
for  some  classes  of  hoisting  work,  or  in  general,work  where  heavy  starting 
torques  and  to  a  certain  extent  variable  speeds  were  required.  The  rotor, 
while  of  the  squirrel-cage  type,  was  made  to  have  a  high  resistance  so 
as  to  give  a  large  starting  torque.  This  rotor  resistance  was  so  adjusted 
that  at  standstill,  about  three -fourths  of  the  maximum  possible  torque 
would  be  developed.  This  would  give  a  fairly  large  value  of  the  torque 
whatever  the  speed. 

The  slip  at  full  load  is  of  course  great.  Moreover,  with  the  large 
slip  the  motor  will  slow  down  nearly  to  standstill  before  reaching  the 
pull  out  point.  These  facts  lead  to  a  convenient  method  of  varying 
the  speed.  This  is  done  by  charging  the  applied  voltage  by  means  of 
an  auto  transformer.  If  with  a  given  torque  the  voltage  is  reduced, 


SPECIAL  TYPE  OF  MOTORS  175 

since  the  flux  is  reduced  in  the  same  proportion,  the  current  must  in- 
crease. This  of  course  leads  to  an  increased  rotor  loss  and  a  con- 
sequently greater  slip. 

The  principal  objection  to  this  method  of  operation  is  the  low  effi- 
ciency on  account  of  the  large  rotor  loss  and  the  great  heating  in  the 
rotor.  On  the  other  hand,  it  is  very  simple  and  would  appear  well 
adapted  to  conditions  where  these  drawbacks  would  be  of  minor  impor- 
tance. 

THE  MILL  TYPE  MOTOR 

Since  the  introduction  of  the  electric  motor  into  steel  mill  practice, 
there  h?s  arisen  a  demand  for  a  motor  of  somewhat  different  character- 
istics than  those  required  for  general  service.  Steel  mill  work  is  typical 
of  a  class  of  power  work  where  everything  must  be  subordinated  to  the 
mechanical  reliability  of  the  motor.  The  great  object  of  the  steel  man 
is  to  get  out  the  tonnage.  The  cost  of  the  power  used,  while  of  course 
a  considerable  item,  is  still  small  in  comparison  with  the  value  of  the 
product.  The  efficiency  of  the  motor  is  therefore  of  minor  importance 
in  comparison  with  its  reliability.  These  considerations  have  led  to 
the  development  of  a  special  and  well  known  form  of  direct-current 
motor,  known  as  the  mill  type. 

In  general,  these  motors  are  entirely  enclosed  so  as  to  prevent  the 
possibility  of  damage  from  the  fall  of  heavy  articles  on  them.  The 
frames  are  usually  of  cast  steel.  The  bearings  are  of  massive  construc- 
tion and  the  size  of  the  spiders,  shafts,  etc.,  is  such  that  the  motor  will 
not  be  injured  by  a  possible  jamming  of  the  rolls  or  other  machinery, 
causing  the  motor  to  be  brought  to  rest  almost  instantly.  In  short, 
every  endeavor  is  made  to  avoid  the  possibility  of  breakdown,  either  on 
account  of  accident  or  ignorant  usage. 

The  demand  for  induction  motors  of  the  mill  type  has  not  been  so 
great  as  for  the  direct-current  motors.  This  is,  of  course,  on  account 
of  the  comparatively  recent  introduction  of  the  use  of  polyphase  alter- 
nating currents  into  service  of  this  kind.  The  greater  simplicity  of 
the  alternating  system,  both  as  regards  the  generators  and  the  motors, 
is  a  powerful  factor  tending  to  increase  the  use  of  alternating  current 
for  just  such  service.  On  the  other  hand,  the  principal  drawbacks  are 
the  fact  that  the  induction  motor  is  not  so  well  adapted  to  variable  speed 
work  as  the  direct-current  motor,  and  the  fact  that  the  starting  torque, 
even  with  the  use  of  a  wound-rotor  machine,  is  by  no  means  so  great  as 
that  of  the  direct-current  series  wound  motor.  In  spite  of  these  unques- 


176  THE  INDUCTION  MOTOR 

tioned  drawbacks,  the  induction  motor  is  gaining  ground,  and  there  is  a 
demand  for  a  motor  in  which  the  strength  of  the  construction  is 
emphasized,  even  with  some  loss  in  the  matter  of  efficiency  and  power- 
factor  and  of  course  with  an  attendant  increase  in  price. 

In  Fig.  86  is  shown  a  motor  of  this  character  built  by  the  General 
Electric  Co.  and  known  as  their  type  M  motor.  At  present,  it  is  built 
for  25-cycle  service  only  and  in  sizes  from  30  to  150  h.p.  As  regards 
strength  of  parts,  and  general  construction,  it  corresponds  to  what  has 
been  said  regarding  this  type  of  machine.  Special  attention  has  been 
given  to  ease  of  repairs,  and  provision  has  been  made  so  that  the  stator 
or  the  rotor  can  be  readily  removed  in  case  of  accident  and  a  spare  part 
substituted.  The  motors  are  built  only  in  the  wound-rotor  type,  and 
every  part,  including  the  slip  rings,  is  completely  enclosed. 


FIG.  86. — General  Electric  Induction  Motor,  Mill  Type. 

The  rating  of  these  motors  is  on  much  the  same  basis  as  that  of 
traction  motors.  They  are  guaranteed  to  carry  full  load  for  one  hour 
with  a  temperature  rise  of  from  65  to  75  degrees  centigrade.  They  will 
also  carry  25  per  cent  overload  for  ten  minutes  without  injurious  heating, 
and  100  per  cent  overload  momentarily.  Their  continuous  rating  is 
25  per  cent  of  their  rated  power. 

It  might  be  pointed  out  that  is  it  particularly  difficult  to  design  an 
induction  motor  of  the  completely  enclosed  type  for  continuous  opera- 
tion, on  account  of  the  necessarily  rather  large  iron  loss  and  the  fact 
that  even  at  no  load  the  stator  current  will  be  about  25  per  cent  of  the 
full  load  current.  It  is  therefore  difficult  to  get  the  losses  low  enough 
so  that  the  machine  can  radiate  the  heat  developed,  even  though  the 
rating  be  made  very  low  for  the  frame  used. 


SPECIAL  TYPE  OF  MOTORS 


177 


THE  WESTINGHOUSE  TYPE  MS  MOTOR 

The  Westinghouse  Company  also  manufacture  a  mill  type  motor 
known  as  the  MS  motor.  Its  general  appearance  is  well  shown  in  Fig. 
87.  The  design  follows  in  general  the  lines  already  indicated  as  regards 
strength,  size  of  parts,  etc.  The  motors  are,  however,  usually  operated 
as  open  motors,  but  can  be  partially  or  entirely  enclosed  if  necessary. 
The  rotors  used  are  of  the  squirrel-cage  variety.  The  peculiar  con- 


FIG.  87. — Westinghouse  Mill  Type  Induction  Motor. 

struction  of  the  end  rings  is  illustrated  in  Fig.  98,  and  is  described  in 
connection  with  the  descriptions  of  various  end  ring  constructions. 


TYPICAL  CONSTRUCTIONS  or  VARIOUS  INDUCTION  MOTOR  PARTS 

A  brief  review  of  some  of  the  various  elements  that  enter  into  the 
construction  of  the  induction  motor  may  be  of  interest.  The  author 
has  attempted  to  select  examples  which  show  in  each  case  the  latest 
practice  of  the  manufacturer,  whose  product  is  described. 

Frames.  The  most  common  type  of  frame  is  perhaps  that  illus- 
trated in  Fig.  88,  which  is  a  view  of  a  standard  Fairbanks  Morse  Induc- 
tion Motor. 


178  THE  INDUCTION  MOTOR 

The  body  of  the  frame  is  usually  constructed  with  numerous  open- 
ings, so  as  to  afford  a  free  exit  for  the  heated  air  from  the  interior  of  the 
motor.  The  bearing  arms  are  frequently  constructed  with  a  view  of 
affording  protection  to  the  stator  winding  where  it  extends  beyond  the 
core.  Bearing  arms  are  in  universal  use  instead  of  the  pedestal  type 
of  frame,  formerly  much  used  for  motors  of  both  the  alternating-current 
and  direct-current  type.  The  difficulty  experienced  in  machining  such 


FIG.  88. — Fairbanks  Morse  Induction  Motor. 

a  frame  together  with  the  added  weight,  has  caused  its  abandonment 
except  in  the  very  largest  sizes  of  motors. 

In  Fig.  89  is  shown  a  type  of  frame  used  by  the  Fort  Wayne  Electric 
Works  in  its  medium  sized  motors.  This  is  known  as  the  skeleton  type 
frame.  As  will  be  readily  seen,  it  is  similar  to  the  frame  just  described 
except  that  the  portion  of  the  iron  between  the  ribs  is  omitted.  This 
results  in  saving  a  great  deal  of  weight,  and  need  not  weaken  the  frame 
if  the  ribs  are  made  sufficiently  strong.  It  is  claimed  that  by  this  con- 
struction, the  laminations  are  directly  exposed  to  the  air,  and  are 
therefore  enabled  to  radiate  their  heat  more  effectively.  While  this 
is  doubtless  true,  it  must  be  remembered,  on  the  other  hand,  that  the 
motor  is  deprived  of  the  radiating  effect  of  that  part  of  the  frame  which 
has  been  cut  away.  The  material  of  the  frame  being  in  direct  contact 


SPECIAL  TYPE  OF  MOTORS  179 

with  the  laminations  becomes  nearly  as  hot  as  the  latter,  and  is  con- 
sequently of  great  value  in  keeping  the  motor  cool.  It  is  a  question 
for  the  manufacturer  to  decide,  whether  or  not  the  amount  saved  by 
using  the  open  type  of  frame  will  if  invested  in  more  laminated  iron 
and  copper,  lead  to  a  motor  that  is,  on  the  whole,  more  satisfactory. 


FIG.  89. — Fort  Wayne  Induction  Motor,  Skeleton  Frame. 

Fig.  90  shows  a  General  Electric  motor  with  what  is  known  as  the 
riveted  type  of  frame.  When  using  this  type  of  construction,  the  lam- 
inations and  the  two  end  plates  are  assembled  in  the  correct  relation, 
and  are  then  bound  together  permanently  by  rivets.  This  form  of  con- 
struction leads  to  a  very  light  motor. 

In  all  of  these  forms,  the  end  brackets  are  held  by  either  four  or 
eight  bolts.  This  allows  the  brackets  to  be  rotated  through  either  90 
or  1 80  degrees,  and  consequently  allows  the  motor  to  be  mounted  on  a 
wall  or  ceiling. 

In  the  case  of  motors  of  large  size,  a  box  frame  with  pedestal  bear- 
ings is  commonly  used.  A  frame  of  this  character,  used  with  a  Westing- 


180 


THE  INDUCTION  MOTOR 


house  motor  of  650  h.p.  is  shown  in  Fig.  91.     In  the  case  of  still  larger 
motors,  the  stator  and  the  pedestals  may  be  mounted  directly  on  the 


FIG.  90. — General  Electric  Motor,  Riveted  Type  Frame. 


FlG.  91. — Heavy  Duty  Westinghouse  Induction  Motor. 

foundations.    A  6000  h.p.  slow  speed  66oo-volt  General  Electric  motor, 
in  which  this  form  of  construction  is  used,  is  shown  in  Fig.  92. 


SPECIAL  TYPE  OF  MOTORS 


181 


The  enclosed  type  of  induction  motor  is  in  less  demand  than  corre- 
sponding sizes  of  direct-current  motors,  since  the  induction  motor  is 
better  able  to  withstand  the  effect  of  dirt,  dampness,  etc.  Occasionally, 
in  case  of  extremely  bad  conditions,  particularly  if  the  load  is  of  an 
intermittent  nature,  they  may  be  desirable.  In  Fig.  93  is  shown  a 
completely  enclosed  General  Electric  motor.  The  annular  radiating 
rings  are  provided  to  assist  in  getting  rid  of  the  heat  generated.  Such 
motors  are  generally  designed  for  operation  with  intermittent  loads. 
If  required  for'  continuous  operation,  it  would  be  necessary  to  design 


FIG.  92.— 6000  H.P.  6600  Volt  General  Electric  Induction  Motor  in  Rail  Mill, 
Gary,  Ind. 

them  with  very  low  flux  densities,  and  consequently  with  small  output 
for  a  given  weight. 

Rotor  Construction.  In  Fig.  94  is  shown  the  rotor  of  a  Westinghouse 
motor.  This  illustrates  a  common  form  of  construction,  using  solid 
end  rings,  fastened  to  the  rotor  bars  by  means  of  bolts.  In  addition, 
to  preserve  the  joint  from  oxidation  and  improve  the  contact,  the  joint 
is  also  soldered.  Instead  of  bolts  to  hold  the  bars  to  the  ring,  rivets  are 
sometimes  used. 

Fig.  95  shows  the  form  of  rotor  construction  used  by  the  Fairbanks 
Morse  Co.  The  bars  are  relatively  thin  and  a  deep  notch  is  formed 
in  both  ends  of  each  bar.  The  bars  and  rings  are  first  assembled  and 


182 


THE  INDUCTION  MOTOR 


FIG.  93.— General  Electric  Motor  Completely  Enclosed. 


FIG.  94.— Squirrel-cage  Rotor  of  Westinghouse  Motor. 


SPECIAL  TYPE  OF  MOTORS 


183 


securely  soldered  by  dipping  each  end  of  the  rotor.  The  rotor  is  then 
placed  in  a  lathe,  the  ends  of  the  rotor  bars  are  turned  off,  and  a  small 
notch  is  formed  on  the  ends  of  the  bars.  A  brass  ring,  having  a  pro- 
jecting lip,  bored  to  the  same  diameter  as  the  notch  just  mentioned, 
is  then  forced  on  each  end  of  the  rotor  bars  and  secured  by  a  number 
of  screws  passing  through  the  retaining  ring  and  into  the  end  ring.  The 
whole  is  then  dipped  in  solder  for  added  strength  and  to  seal  the  ends 
of  the  screws,  and  is  then  trued  up  in  the  lathe. 

The  type  of  construction  used  by  the  Burke  Electric  Co.,  in  which 
there  are  no  joints  in  the  rotor  conductors,  has  already  been  described 
on  page  173. 


FIG.  95. — Construction  of  Fairbanks-Morse  Squirrel-cage  Rotor. 

A  type  of  soldered  end  ring  largely  used  by  the  General  Electric  Co., 
is  shown  in  Fig.  96.  As  will  be  apparent  from  the  illustration,  each 
ring  is  made  up  of  several  parts.  This  affords  excellent  facilities  for 
ventilation,  when  the  rotor  is  in  operation,  and  the  cross-section  of 
metal  can  in  consequence  be  small.  While  the  motor  is  at  rest  or  is 
revolving  slowly,  the  rings  are  only  slightly  cooled  by  the  fanning  action 
of  the  air,  and  they  consequently  become  much  hotter  than  would  solid 
rings  of  a  correspondingly  larger  cross-section.  The  rings  may  be 
made  of  a  material  having  a  large  temperature  coefficient.  They  will 


184 


THE  INDUCTION  MOTOR 


consequently  increase  considerably  in  resistance  during  the  starting 
period,  thus  giving  a  good  starting  torque,  but  will  quickly  cool  off 
as  soon  as  the  rotor  reaches  full  speed.  They  will  thus  act  automatically 
to  give  greater  starting  torque  than  would  be  possible  with  the  same 
slip  with  the  ordinary  type  of  solid  end  ring  construction.  It  is  of  course 
evident  that  care  must  be  taken  to  keep  this  rise  of  temperature  within 
suitable  limits,  as  otherwise  the  solder  might  be  melted  from  the  rings. 


FIG.  96. — General  Electric  Squirrel-cage  Rotor. 


In  Fig.  97  is  shown  the  rotor  of  a  Crocker- Wheeler  motor  of  some- 
what similar  design.  In  this,  however,  the  rings  instead  of  being  con- 
tinuous around  the  whole  circle,  consist  of  short  segments.  In  assem- 
bling these  segments,  they  are  placed  on  the  ends  of  the  rotor  bars  in 
such  a  manner  that  each  one  overlaps  the  one  ahead  of  it.  Looking 
down  on  the  rotor,  the  rings  present  a  spiral  appearance.  It  is  claimed 
for  this  method  of  construction  that  it  obviates  the  tendency  present 
in  the  form  previously  described  for  the  inner  ring  to  take  more  than 
its  share  of  the  current.  In  this  way,  the  danger  of  the  inner  ring  becom- 
ing much  hotter  than  the  outer  ones  is  avoided. 


SPECIAL  TYPES  OF  MOTORS 


185 


In  Fig.  98  is  shown  a  form  of  end  ring  used  by  the  Westinghouse  Co. 
on  its  mill  type  motors.  Their  machines  are  designed  to  withstand 
the  roughest  sort  of  usage,  and  in  consequence  every  effort  is  made  to 


FIG.  97. — Squirrel-cage  Rotor,  Crocker- Wheeler  Induction  Motor. 

make  the  construction  as  simple  and  durable  as  possible.     As  shown 
in  the  illustration,  the  end  rings  consist  of  punched  sheets  of  resistance 


a-Kotcw  spider  b-  Laminated  core  C  -Conductor  d-Bciistauce  rings. 6  -End  ring,  f  -Key.g-Lock  washer 

FIG.  98. — Construction  of  Rotor  of  Westinghouse  Mill  Type  Motor. 


metal,  and  are  provided  with  lips  turned  at  right  angles,  and  adapted 
to  make  contact  with  the  rotor  bars.  They  are  secured  to  the  latter 
as  shown,  and  are  further  supported  by  insulated  studs,  screwed  into 
the  rotor  spider.  No  solder  is  used. 


186  THE  INDUCTION  MOTOR 

GENERAL  NOTES  ON  THE  SELECTION  OF  MOTORS.    SPEED 

From  the  standpoint  of  the  motor  itself,  the  higher  the  speed  the 
better.  Of  course  in  the  majority  of  cases,  the  speed  is  fixed,  at  least  to 
a  certain  extent,  by  considerations  external  to  the  motor.  If  these  are 
such  that  the  speed  of  the  motor  must  be  fixed  at  some  rather  low  value, 
the  manufacturer  is  of  course  compelled  to  do  the  best  he  can  under  the 
circumstances.  It  must,  however,  be  kept  in  mind  that  the  motor  for 
the  lower  speed  will  be  higher  in  price,  and  at  the  same  time  lower  in 
efficiency,  power-factor,  pull  out  point,  and  starting  torque. 

On  account  of  the  lower  speed,  a  larger  frama  must  of  course  be 
used,  thus  increasing  the  cost.  Also,  since  the  amount  of  .active  mater- 
ial is  greater,  if  as  is  usually  the  case,  this  active  material  is  worked  at 
the  same  flux  densities  and  current  densities,  these  losses  will  be  greater 
or  the  efficiency  will  be  lower. 

The  fact  that  the  power-factor  of  any  induction  motor  is  less  than 
unity,  is  due  to  the  fact  that  the  motor  requires  a  current  lagging  90 
degrees  behind  the  e.m.f.  to  maintain  the  magnetic  flux  across  the  air 
gap  and  through  the  iron.  The  value  of  the  magnetomotive  force  is 
proportional  to  the  length  of  the  air  gap,  and  the  number  of  poles. 
It  is  entirely  independent  of  the  size  of  each  of  the  poles.  The  slow- 
speed  motor  requires  a  greater  number  of  poles  than  one  of  higher  speed, 
and  consequently  requires  a  greater  magnetomotive  force,  and  has  a 
correspondingly  lower  power-factor.  To  overcome  this  to  a  certain 
extent,  the  manufacturer  is  forced  to  use  the  shortest  possible  air  gap, 
resulting  in  a  motor  which  will  not  withstand  so  much  wear  before  the 
rotor  will  come  into  contact  with  the  stator,  as  would  have  been  the 
case  if  a  more  conservative  design  could  have  been  adopted. 

This  objection  has  far  more  weight  in  the  case  of  6o-cycle  motors 
than  in  the  case  of  those  for  25  cycles,  since  in  the  latter  the  number  of 
poles  for  a  given  speed  is  far  less  than  in  a  6o-cycle  machine. 

FREQUENCY 

The  question  of  the  frequency  to  be  employed  is  in  the  majority 
of  cases  settled  beforehand  by  other  considerations.  In  any  event  the 
question  has  been  fully  discussed  in  Chapter  IX,  and  we  may  merely 
make  the  statement  that  in  general,  taking  everything  into  consideration, 
a  frequency  of  60  cycles  is  the  best  when  the  majority  of  the  motors  are 
to  be  of  small  and  medium  size,  and  that  25  cycles  is  the  better  when  the 
motors  average  about  100  h.p.  or  over. 


SPECIAL  TYPES  OF  MOTORS  187 

SIZE  OF  MOTORS 

Every  effort  should  be  made  to  select  the  smallest  possible  motor 
for  the  work  in  hand.  It  is  not  intended  to  imply  by  this  that  motors 
so  small  that  they  will  be  regularly  overloaded  should  be  used,  bu 
that  for  example  a  5o-h.p.  motor  should  not  be  used,  if  a  35-h.p.  machine 
is  of  ample  size  to  carry  the  load.  In  direct-current  work  it  is  often 
the  best  practice  to  use  large  motors,  in  the  expectation  that  future  growth 
will  so  increase  the  load  on  the  motors  as  to  render  them  ultimately  of 
the  correct  size.  With  direct  currents,  this  merely  involves  a  somewhat 
greater  initial  investment  and  a  slightly  lower  efficiency  as  long  as  the 
load  is  light.  To  offset  this  it  is  argued  that  in  addition  to  the  motors 
being  ultimately  of  the  correct  size,  there  will  in  the  meantime  be  less 
liability  of  trouble  since  the  motors  will  be  of  excess  size,  and  it  will 
not  be  necessary  to  interrupt  production  to  change  to  a  larger  size. 

Of  course  no  one  can  deny  that  these  arguments  have  weight  also 
in  the  case  of  an  alternating-current  installation.  The  fact  that  induc- 
tion motors  have  a  relatively  low  power-factor  when  operated  at  partial 
loads,  should  act  powerfully  to  prevent  the  use  of  a  larger  motor  than 
is  necessary.  As  a  matter  of  fact,  the  best  power-factor  is  usually  found 
between  100  and  125  per  cent  of  full  load,  and  in  some  cases,  may  be 
found  at  an  even  larger  overload. 

The  bad  results  attributed  to  the  use  of  underloaded  induction 
motors  operating  at  low  power-factors,  are  noted,  not  so  much  in  the 
motor  itself,  as  they  are  in  the  generator  supplying  the  power.  Low 
power-factor  results  both  in  loading  the  generator  up  to  its  full  current 
capacity  long  before  its  power  limit  is  reached,  but  what  is  perhaps 
more  serious,  acts  powerfully  to  spoil  the  regulation.  This  is  of  course 
more  apparent  in  cases  where  the  generator  power  rating  is  relatively 
small  compared  with  that  of  the  induction  motors  fed  by  it.  For 
example,  if  half  the  full  load  generator  rating  were  required  for  incan- 
descent lamps,  and  the  other  half  by  induction  motors,  the  effects  of  the 
poor  regulation  caused  by  lightly  loaded  motors  would  not  be  so  apparent 
as  would  be  the  case  if  all  the  power  were  required  by  induction  motors. 

There  are  several  ways  in  which  this  poor  regulation  shows  itself. 
If  lamps  are  fed  from  the  same  generator,  the  variations  of  the  voltage 
will  result  in  very  poor  service.  This  will  be  especially  the  case  if  the 
lamps  are  on  the  same  circuit  as  the  motors.  As  regards  the  motors, 
the  effects  will  be  apparent  in  insufficient  starting  torque,  and  a  tendency 
of  the  motors  to  pull  out  on  overloads  which  they  would  readily  carry 


188  THE  INDUCTION  MOTOR 

if  the  voltage  were  normal.  Since  both  the  starting  torque  and  the  pull 
out  point  are  proportional  to  the  square  of  the  voltage,  this  may  easily 
become  serious.  Moreover,  the  trouble  is  cumulative.  That  is,  the 
low  power-factor  causes  the  voltage  to  drop.  This  would  of  itself 
call  for  a  proportionately  larger  current  to  supply  the  same  power  through 
the  motors,  but  in  addition,  on  account  of  the  lowered  voltage,  the  motor 
is  virtually  changed  to  a  machine  of  smaller  rating,  and  on  account 
of  the  large  load,  it  may  work  at  still  lower  power-factor.  This  in  turn 
again  reduces  the  voltage,  and  so  on.  Thus  it  will  be  seen  that  trouble 
of  this  sort  once  started  may  readily  grow,  and  even  become  so  bad 
as  to  render  the  operation  of  the  plant  at  its  rated  load  entirely  imprac- 
ticable. 

Of  course,  this  condition  of  affairs  may  be  helped  to  some  extent 
by  the  installation  of  an  automatic  regulator,  but  on  account  of  the 
fact  that  it  requires  a  short  time  for  the  regulator  to  act,  it  may  not 
be  as  useful  in  overcoming  the  difficulty  as  might  at  first  sight  appear 
probable.  The  safest  course  is  to  install  generators  of  ample  size  and 
hence  of  good  regulation,  and  motors  of  the  smallest  size  possible  with 
the  load  to  be  carried.  If  in  the  future  it  is  necessary  to  use  motors  of 
higher  rating,  this  may  frequently  be  done  without  great  expense  or 
inconvenience  by  interchanging  the  heavily  loaded  motors  for  others 
which  are  of  larger  size  (using  other  motors  already  installed)  and  in 
turn  substituting  others  for  those  used  in  this  way. 

SQUIRREL-CAGE  AND  WOUND-ROTOR  MACHINES 

In  every  installation  the  question  of  whether  to  employ  wound- 
rotor  machines  or  those  with  squirrel-cage  rotors  must  be  determined. 
In  general  we  may  say  that  the  wound-rotor  machine  is  a  necessary 
evil.  For  operation  at  full  speed  and  after  the  machine  is  in  motion,  the 
squirrel-cage  machine  is  in  every  respect  superior.  The  one  possible 
exception  is  in  the  matter  of  efficiency.  If  the  squirrel-cage  motor  is 
required  to  develop  starting  torque  with  a  limited  starting  current, 
it  is  necessary  to  use  a  rotor  of  high  resistance,  and  this  may  in  certain 
cases  reduce  the  efficiency  to  that  of  a  wound-rotor  machine  or  even  to 
a  lower  value.  In  regard  to  price,  power-factor,  pull  out  point,  rug- 
gedness  of  construction,  and  freedom  from  liability  of  trouble,  the 
squirrel-cage  machine  has  an  unquestioned  advantage. 

It  is  in  the  matter  of  starting  torque  and  availability  for  variable 
speed  work  that  the  wound-rotor  machine  has  an  advantage.  To 


SPECIAL  TYPES  OF  MOTORS  189 

develop  100  per  cent  starting  torque  with  a  squirrel-cage  machine  will 
require  approximately  300  to  400  per  cent  of  full-load  current,  and  'this 
current  is  unfortunately  lagging  almost  90  degrees  behind  the  applied 
e.m.f.  As  a  consequence,  the  voltage  regulation  of  the  circuit  is 
seriously  interfered  with,  and  it  is  usually  necessary  to  take  the  starting 
current  from  back  of  the  fuses,  thus  leaving  the  motor  without  protec- 
tion during  the  starting  period.  On  the  other  hand,  the  wound-rotor 
machine  will  develop  100  per  cent  torque  with  but  little  if  any  more 
than  100  per  cent  of  full-load  current.  Other  torques  require  current 
in  proportion.  Moreover,  this  current  is  of  a  much  higher  power-factor 
than  that  of  the  squirrel-cage  motor  during  starting,  and  hence  the  line 
disturbance  is  even  less  in  proportion  than  would  be  indicated  by  the 
ratio  of  the  currents. 

This  large  starting  current  of  the  squirrel-cage  motor  may  or  may 
not  be  of  importance,  depending  upon  various  conditions  external  to 
the  motor.  Thus  if  the  generator  supplying  the  energy  be  of  large  size 
in  comparison  with  the  motors  to  be  started,  the  disturbance  at  starting 
will  be  small.  In  order  that  this  may  generally  be  true,  the  generator 
should  be  capable  of  furnishing  approximately  ten  times  the  full  load 
current  of  the  largest  motor.  With  such  a  proportion  the  starting  of 
an  induction  motor  of  the  squirrel-cage  type  should  cause  a  fall  of 
voltage  of  not  much  more  than  10  per  cent. 

If  the  motors  are  larger  relatively  to  the  generator  than  the  figures 
just  given,  say  the  generator  is  capable  of  furnishing  twice  the  full- 
load  current  of  the  motor,  we  may  still  secure  satisfactory  service  pro- 
viding the  load  at  starting  can  be  made  very  light,  or  providing  the 
large  fall  in  voltage  during  starting  is  not  objectionable.  For  example, 
it  might  be  desirable  to  start  a  large  motor  of  this  character  only  in  the 
morning"  and  at  the  noon  hour.  It  would  be  entirely  practicable  to 
start  this  motor  first  and  then  start  the  smaller  motors  or  other  devices 
when  convenient.  The  large  fall  in  voltage  would  then  not  affect  any 
other  apparatus  on  the  line. 

The  ratio  given  above,  that  is  the  k.v.  amp.  rating  of  the  genera- 
tor double  the  h.p.  rating  of  the  motor,  is  about  the  limiting  ratio  that 
can  be  employed  at  all,  if  it  is  necessary  to  develop  full-load  torque 
during  starting.  To  do  this  requires  that  an  amount  of  power  equal  to 
the  full  h.p.  rating  of  the  motor  be  wasted  in  the  rotor.  In  order  to 
get  the  current  corresponding  to  this  loss  into  the  rotor  requires  a  loss 
in  the  stator  of  nearly  the  same  amount.  In  some  cases  this  stator  loss 
may  be  even  greater  than  the  rotor  l^ss,  although  in  standard  motors 


190  THE  INDUCTION  MOTOR 

it  will  in  general  be  less.  On  the  average  we  shall  not  be  far  wrong 
if  we  assume  that  to  start  a  squirrel-cage  motor  under  full-load  torque 
requires  approximately  twice  full-load  power  to  be  furnished  to  the 
motor.  Hence  in  the  assumed  case,  the  full  k.w.  output  of  the  generator 
would  be  required.  Moreover,  this  power  is  supplied  at  a  low  power- 
factor  and  hence  the  full-load  current  rating  of  the  generator  would  be 
greatly  exceeded. 

If  it  is  desired  to  install  a  generator  and  a  motor  to  take  the  entire 
output  of  the  generator,  it  will  be  necessary  to  employ  a  wound-rotor 
machine  or  to  make  arrangements  so  that  the  motor  can  be  started 
with  little  or  no  load.  On  the  other  hand,  motors  with  phase-wound 
rotors  may  be  freely  employed  up  to  the  rull  rating  of  the  generator. 

These  facts  in  regard  to  the  better  starting  characteristics  of  the 
wound-rotor  machine,  indicate  that  it  should  be  used  in  all  cases  where 
starting  is  of  frequent  enough  occurrence  to  constitute  an  appreciable 
proportion  of  the  whole  running  time.  Such  applications  are  electric 
elevators,  electric  hoists,  crane  motors,  motors  for  electric  traction,  etc. 

Some  of  the  above  involve  work  which  requires  variable  speed. 
For  this  class  of  work  the  squirrel-cage  motor  is  out  of  the  question, 
and  we  are  forced  to  employ  the  wound-rotor  machine.  Unfortunately, 
we  have  nothing  on  the  market  at  the  present  time  that  at  all  corresponds 
to  the  adjustable-speed,  direct-current  shunt  motor.  With  this  type 
of  machine,  it  is  possible  to  set  the  speed  at  a  certain  value  by  changing 
the  shunt  field  current,  and  after  this  adjustment  the  speed  will  be  but 
little  changed  by  subsequent  changes  in  load.  We  have  available  at  the 
present  time  only  one  alternating-current  machine  that  possesses  at 
all  these  characteristics.  This  is  the  induction  repulsion  type  single- 
phase  induction  motor.  Even  this  machine  possesses  these  character- 
istics to  only  a  slight  degree  compared  with  a  direct-current  motor, 
that  is,  the  range  of  speed  adjustment  is  small.  There  is  little  or  no 
doubt  that  a  polyphase  motor  having  the  same  characteristics  could 
be  developed,  and  undoubtedly  such  a  machine  will  be  offered  as  soon 
as  there  is  a  sufficient  demand  for  it.  It  would  have  a  commutator 
and  brushes,  and  means  would  be  provided  for  impressing  various 
voltages  on  the  commutator.  In  addition  to  its  variable  speed  features, 
such  a  motor  could  be  made  to  have  a  power-factor  of  nearly  unity, 
or  even  to  take  a  leading  current.  (See  page  86). 

Unfortunately,  the  slip  ring  type  of  motor  does  not  possess  these 
features;  that  is,  as  the  load  is  varied,  the  speed  will  vary  within  wide 
limits,  particularly  if  the  resistance  of  the  rotor  circuit  is  considerably 


SPECIAL  TYPES  OF  MOTORS  191 

increased  to  bring  the  speed  down  to  a  fraction  of  synchronism.  If 
this  is  the  case  and  the  load  is  removed,  the  speed  will  increase  to  nearly 
synchronism.  It  is  therefore  necessary  to  keep  this  fact  in  mind  in 
using  the  wound-rotor  type  of  induction  motor  for  variable  speed  work. 
This  fact  forces  the  employment  of  direct  current  in  order  to  be  able 
to  utilize  adjustable  speed  motors.  In  many  cases,  however,  it  is 
possible  by  the  use  of  mechanical  speed  changing  devices  to  get  rid  of 
the  necessity  of  the  variable-speed  motors. 

GROUP  OR  INDIVIDUAL  DRIVE 

The  choice  of  the  type  of  motors  to  be  used  in  any  installation  is 
bound  up  with  the  question  of  the  use  of  the  group  drive  or  of  indi- 
vidual drive.  The  group  drive  is  to  a  certain  extent  an  outgrowth  of 
the  older  system  of  driving  through  a  line  shaft.  When  such  a  system 
is  to  be  changed  to  an  electric  drive,  the  most  obvious  and  simple  way 
of  making  the  change  is  to  split  the  shafting  up  into  such  sections  as 
is  most  convenient,  and  drive  each  of  these  sections  by  means  of  a  com- 
paratively large  motor.  In  addition  to  the  fact  that  this  method  reqiures 
the  minimum  amount  of  change  from  the  older  system  of  drive,  it  has 
several  advantages  of  its  own.  The  total  horse-power  of  the  motors 
installed  will  be  much  less  than  would  be  required  if  each  tool  were  to 
be  provided  with  a  motor  of  its  own.  This  condition  arises  largely 
from  the  fact  that  we  can  in  general  count  on  not  having  all  the  machines 
connected  with  a  given  motor  operating  at  the  same  time.  Thus  the 
total  h.p.  output  required  at  any  given  time  will  be  less  than  would 
be  the  case  if  each  tool  had  its  own  motor,  since  in  this  case  it  would 
obviously  be  necessary  that  the  total  rating  of  the  motors  be  equal  to 
the  total  rating  of  the  machine  tools. 

On  the  other  hand,  it  may  frequently  happen  that  it  is  necessary 
to  operate  only  one  small  machine  out  of  the  entire  group.  To  do 
this  it  will  of  course  be  necessary  to  operate  the  motor  connected  to 
the  group.  Since  the  efficiency  of  a  motor  at  light  load  is  small,  this 
results  in  a  waste  of  power. 

A  typical  case  of  a  condition  where  a  group  drive  would  be  indicated 
is  that  of  a  number  of  sewing  machines.  These  are  usually  all  of 
the  same  size,  and  the  nature  of  the  work  is  in  most  cases  such  that 
all  of  the  machines  are  in  operation  at  the  same  time.  The  cost  of 
the  one  motor  to  drive  the  group  is  obviously  less  than  would  be  the 
cost  of  individual  motors  for  each  machine.  The  efficiency  is  also 


192  THE  INDUCTION  MOTOR 

much  higher.  Other  examples,  such  as  groups  of  small  speed  lathes, 
automatic  screw  machines,  etc.,  will  readily  suggest  themselves. 

On  the  other  hand,  a  case  calling  for  the  use  of  individual  drive 
would  be  that  of  a  large  boring  mill,  planer,  or  lathe,  which  was  in 
rather  intermittant  use.  To  operate  a  number  of  machines  of  this 
character  using  group  drive,  would  obviously  require  that  a  long  section 
of  shafting  be  kept  in  motion  almost  all  of  the  time.  This  would  of 
course  result  in  a  large  loss  of  power,  both  in  the  shafting,  and  on  account 
of  the  motor  being  frequently  under-loaded. 

In  case  the  group  drive  is  the  one  selected,  the  motor  to  be  chosen 
will  almost  invariably  be  of  the  squirrel-cage  type.  This  is  evidently 
the  case,  since  the  service  does  not  require  a  large  starting  torque,  and 
the  speed  is  constant.  If  the  amount  of  shafting  used  is  so  long  that 
a  large  starting  torque  is  necessary,  it  would  in  most  cases  be  prefer- 
able to  use  shorter  sections  so  as  to  reduce  this  loss,  or  to  go  to  the 
individual  drive. 

If  individual  drive  is  chosen,  the  best  motor  in  the  majority  of  cases 
will  still  be  the  one  with  squirrel-cage  rotor.  The  tools  can  almost 
invariably  be  started  up  without  load  and  consequently  the  starting 
torque  will  be  small.  If  variable  speed  is  required,  the  wound-rotor 
machine  must  be  used,  but  it  must  also  be  kept  in  mind  that,  as  was 
pointed  out,  it  is  not  possible  to  adjust  a  motor  for  a  given  speed  and 
expect  it  to  retain  that  speed  no  matter  what  the  load  may  be.  Thus  if 
a  lathe  were  equipped  with  a  motor  of  this  character  and  it  were  operat- 
ing at  low  speed  with  a  deep  cut,  and  the  tool  should  run  out  of  the 
cut,  the  motor  would  speed  up  with  disastrous  consequences  when  the 
tool  again  entered  the  cut.  In  the  machine  shop,  as  far  as  the  driving 
of  tools  is  concerned,  there  would  therefore  be  little  use  for  the  wound- 
rotor  machine. 

For  the  operation  of  electric  elevators,  for  crane  motors,  and  for 
hoists,  the  phase-wound  machine  is  the  only  suitable  one.  This  arises 
on  account  of  the  frequent  starting  and  the  large  torque  required  in 
these  applications.  In  most  cases,  also,  these  motors  would  be  rated 
for  intermittent  service. 

MOTORS  FOR  CENTRIFUGAL  PUMPS 

The  application  of  induction  motors  to  centrifugal  pumps  is  becom- 
ing of  very  frequent  occurrence.  For  this  service  the  squirrel-cage 
motor  is  the  one  almost  invariably  recommended.  This  is  certainly 


SPECIAL  TYPES   OF  MOTORS  193 

the  correct  motor  to  use  if  the  head  under  which  the  pump  is  to  work 
is  constant  at  all  times.  If,  however,  as  is  frequently  the  case,  the  head 
varies  from  time  to  time,  the  wound-rotor  machine  should  be  used. 
This  is  on  account  of  the  fact  that  the  speed  of  a  centrifugal  pump  should 
be  rather  accurately  adjusted  to  the  head  under  which  the  pump  is 
operating.  If  this  is  not  done,  the  pump  is  very  inefficient.  It  might 
be  argued  that  we  might  as  well  waste  the  power  in  the  pump  as  in  the 
regulating  resistance  of  the  motor,  but  it  can  readily  be  shown  that  less 
power  input  to  the  motor  will  be  required  if  the  motor  and  pump  are 
slowed  down  by  the  increase  of  resistance  in  the  rotor  circuit  than  would 
be  the  case  if  the  machine  were  allowed  to  operate  at  full  speed.  This 
is  true  even  in  spite  of  the  loss  in  the  regulating  rheostat. 


CHAPTER  XIII 

THE  SINGLE-PHASE  INDUCTION  MOTOR 

IN  the  preceding  pages  we  have  shown  that  in  a  polyphase  motor 
with  an  infinite  number  of  phases,  if  each  of  these  phases  is  supplied 
with  an  e.m.f.  of  the  proper  magnitude  and  phase,  and  if  the  applied 
e.m.f .  is  sinusoidal,  a  sinusoidal  band  of  flux  rotating  with  synchronous 
speed  will  be  set  up.  The  same  result  will  also  follow  if  we  have  only  a 
finite  number  of  phases,  say  two  or  three,  provided  the  conductors 
of  each  phase  are  distributed  in  proportion  to  the  ordinates  of  a  curve  of 
sines.  If  this  is  the  case,  the  sheet  of  current  set  up  by  the  current 
of  each  phase  is  harmonic.  In  a  practical  motor,  it  is  impracticable 
to  distribute  the  conductors  in  this  manner,  and  we  frequently  content 
ourselves  with  arranging  the  conductors  and  consequently  the  current 
in  bands  of  uniform  strength.  In  the  single-phase  motor,  however,  some 
attempt  is  often  made  to  approximate  a  sinusoidal  distribution. 
Frequently  a  short-pitch  winding  is  used  in  polyphase  motors,  and  this 
has  the  effect  of  making  the  bands  overlap  and  consequently  the 
current  approaches  more  nearly  to  an  harmonic  distribution.  In 
either  case  the  tendency  is  to  set  up  a  non-harmonic  band  of  flux,  as 
shown  in  Figs.  8  and  9.  The  currents  produced  in  the  rotor  conduc- 
tors, particularly  in  the  case  of  a  squirrel-cage  rotor,  act  powerfully, 
however,  to  prevent  any  change  of  flux,  and  the  actual  rotating  mag- 
netic field  is  very  nearly  harmonic. 

The  single-phase  induction  motor  may  be  considered  as  a  special 
case  of  the  polyphase  motor.  If  a  two-phase  motor  is  operating  without 
load  and  one  of  the  phases  is  opened,  the  motor  continues  to  run  at 
almost  exactly  the  same  speed  as  before.  The  only  apparent  change 
is  that  the  nature  of  the  hum  emitted  by  the  motor  changes  slightly 
in  character.  If  an  ammeter  is  used  to  measure  the  current  taken  by  the 
machine  (in  this  case  principally  magnetizing  current)  it  will  be  found 
that  the  current  in  the  phase  still  connected  has  approximately  doubled, 
as  has  likewise  the  power  taken  by  this  phase.  The  total  current  and 
the  total  power  are  nearly  unchanged. 

194 


THE  SINGLE-PHASE  INDUCTION  MOTOR  195 

If  when  operating  in  this  manner,  a  voltmeter  be  applied  to  the 
idle  phase  it  will  be  found  that  nearly  the  full  line  voltage  is  present 
there,  and  a  further  investigation  will  show  that  this  e.m.f.  differs  90 
degrees  in  phase  from  the  line  voltage.  The  application  of  test  coils 
at  various  angles  with  the  active  phase  would  show  the  presence  of 
nearly  the  same  e.m.f.  regardless  of  the  position  of  the  coil,  and  an 
angular  difference  of  phase  corresponding  to  the  angle  of  the  coil  with 
the  stator  winding.  This  experiment  proves  the  existence  of  a  rotating 
magnetic  field,  and  a  fuller  investigation  would  show  that  this  field 
is  harmonic  in  its  space  distribution  and  rotates  with  uniform  angular 
velocity. 

In  the  light  of  what  has  previously  been  said,  it  is  almost  self- 
evident  that  this  will  be  the  case.  The  flux  tends  to  assume  such  a 
distribution  and  value  that  the  minimum  cutting  of  the  rotor  conductors 
and  consequently  the  minimum  expenditure  of  power  will  take  place. 
Each  change  of  flux  sets  up  a  rotor  current  in  such  a  position  and  phase 
as  to  tend  to  prevent  the  change  of  flux.  With  a  rotor  operating  at 
synchronism,  a  uniform  flux  rotating  at  uniform  velocity  would  not 
cut  the  rotor  bars  at  all.  Hence  the  flux  tends  to  assume  this  distribu- 
tion and  velocity. 

However,  to  produce  the  rotating  magnetic  field,  in  a  single-phase 
induction  motor  it  is  evident  that  there  must  exist  a  component  of  m.m.f. 
at  right  angles  to  the  axis  of  the  stator  winding.  Since  the  stator  can 
not  carry  a  current  in  the  proper  position  to  produce  this  it  must  exist 
in  the  rotor.  To  produce  this  rotor  current  requires  a  change  in  the 
stator  flux.  Hence  the  field  can  not  be  of  absolutely  constant  value 
at  all  times.  The  change  however  is  slight. 

The  nature  of  the  current  required  in  the  rotor  to  produce  with 
the  stator  current  a  rotating  magnetic  field  in  the  motor,  will  be  apparent 
from  Fig.  99.  The  four  diagrams  are  drawn  for  successive  values  of 
the  current  taken  at  intervals  of  30  degrees.  The  flux  is  shown  dis- 
placed 30  degrees  to  the  left  for  each  change  of  30  degreesin  the  current. 

The  solid  line  marked  stator  current  represents  the  distribution  of 
the  stator  current  over  the  stator  surface.  The  conductors  are  assumed 
to  be  arranged  on  the  stator  core  in  such  a  manner  that  the  number 
of  conductors  at  any  given  point  is  proportional  to  the  sine  of  the  angle 
corresponding  to  the  point.  The  arc  between  two  poles  is  of  course 
taken  as  180  electrical  space  degrees.  The  current  is  the  same  in  all 
of  the  conductors  and  hence  the  sinusoidal  curve  represents  the  distribu- 
tion of  the  current  over  the  stator  core.  The  curve  of  distribution  is 


196 


THE  INDUCTION  MOTOR 


stationary  in  space,  but  variable  in  magnitude,  changing  from  a  positive 
to  a  negative  maximum,  in  accordance  with  the  change  in  the  current. 
In  order  that  an  harmonic,  uniformly  rotating  magnetic  flux  be 


One  Pole          One  Pole 


0  Amp. 


) 

>^ 

\F 
\ 

Z 

^/ 

\ 

\ 

( 

j~ 

^ 

K 

/ 

V 

:1 

F=  Flux 
S  =  Stator  Current 
R=  Rotor  Current 
T—  Resultant  Current 

FIG.  99. — Current  Distribution  in  Stator  .and  Rotor  of  a  Single-phase  Induction 
Motor.     Sinnusoidal  Distribution  of  Stator  Coils,  Rotor  Squirrel-cage. 

maintained,  it  is  necessary  that  a  resultant  harmonic  band  of  current 
rotate  uniformly  around  the  stator.  This  resultant  is  due  to  currents 
in  both  the  stator  and  the  rotor.  It  is  shown  as  a  dotted  line  in  each 
of  the  figures  and  is  drawn  90  degrees  ahead  of  the  flux.  The  resultant 


THE  SINGLE-PHASE  INDUCTION  MOTOR  197 

band  of  current  is  due  to  the  algebraic  sum  of  the  current  sheets  in 
both  the  stator  and  rotor  at  any  given  point.  The  rotor  current  is  then 
the  difference  between  the  resultant  current  and  the  stator  current.  It 
is  shown  by  the  curve  marked  rotor  current.  In  order  that  the  required 
rotor  current  may  circulate,  it  is  necessary  that  the  rotor  be  of  the  squirrel- 
cage  variety  with  many  bars.  With  a  phase-wound  rotor,  the  current 
could  not  assume  the  exact  values  required  at  all  points  and  the  result- 
ing rotating  flux  would  not  have  exactly  harmonic  distribution. 

A  study  of  the  construction  of  the  diagrams  will  reveal  the  following 
facts: 

A.  The  flux  has  harmonic  distribution,  is  constant  in  magnitude 
and  rotates  uniformly  in  the  direction  of  rotation  of  the  rotor. 

B.  The   stator  current  sheet  is  stationary  in  space  distribution, 
and  has  harmonic  variation  in  magnitude. 

C.  The  rotor  current  sheet  is  harmonic  in  space  distribution,  of 
constant  maximum  value,  and  rotates  backward,  i.e.,  opposite  to  the 
direction  of  rotation  of  the  rotor  at  synchronous  speed.     Its  maximum 
value  is  half  that  of  the  stator  current  sheet. 

It  is  apparent  that  at  the  time  shown  in  Figure  D,  the  rotor  current 
sheet 'must  be  sufficient  to  force  the  total  flux  across  the  gap.  Hence  its 
value  is  the  same  as  would  be  required  in  a  second  phase  of  the  stator  if 
one  were  present.  Its  space  location  is  90  degrees  from  the  stator  wind- 
ing. The  conclusion  readily  follows,  that  a  single-phase  motor  requires 
twice  the  magnitizing  current  that  would  be  taken  by  one  phase  of  the 
same  motor- wound  two-phase  with  the  same  number  of  turns  per  phase. 
Similar  reasoning  would  apply  to  a  three-phase  motor  compared  with 
a  single-phase,  or  in  general  we  may  say  that  the  volt-amperes  required 
are  the  same  whatever  the  number  of  phases.  The  same  principle 
was  explained  in  developing  the  formula  for  the  magnetizing  current 
of  a  polyphase  motor.  The  foregoing  may  be  considered  as  a  proof 
that  the  same  formula  applies  to  the  single-phase  motor. 

The  curves  of  Fig.  99  are  constructed  on  the  supposition  that  the 
conductors  on  the  stator  core  are  distributed  in  such  a  manner  that 
the  number  of  conductors  at  any  given  point  is  proportional  to  the 
sine  of  the  angle  at  that  point,  counting  from  some  fixed  point  of  refer- 
ence on  the  stator  core.  It  is  hardly  necessary  to  point  out  that  such 
a  distribution  is  not  feasable  in  practice.  In  many  motors,  some  attempt 
is  made  to  approximate  this  condition  by  winding  some  coils  with 
less  turns  than  others.  To  do  this  requires  coils  of  the  concentric 


198 


THE  INDUCTION  MOTOR 


type,  as  shown  in  Fig.  79.     For  this  reason,  and  for  the  sake  of  sym- 
metry, such  coils  are  frequently  employed  in  single-phase  motors. 


a-  Stator  Current 

/     "\ 

•»                   / 

.1. 

/ 

\ 

\ 

7 

tor  Current— 

^/""\ 

\ 

^ 

/    Resultant 
v  Current 

r              >vi 

^-'' 

^^ 

v<*-  Rotor  Current 

s=».866 

/' 

\ 

s 

V 

/ 

\ 

\ 

\ 

/ 

\ 

•       •• 

v 

\ 
\ 

v         / 

\ 

/ 

1    / 

\ 

/       T»               1J. 

I  / 
:/  Stator  Current-"* 

\ 

Sf~~  Resultan 
Current 

Kesultant  Current 


Rotor  Current 


/      / 

/    /     Stator 
A      ,•       Current 


Rotor  Current~-y'                      N^-Resultant  Current 
0.  /  \ 


FIG.  100. — Current  Distribution  in  Stator  and  Rotor  of  Single-phase  Induction 
Motor.     Stator  Coils  in  120  Degree  Bands.     Rotor  Squirrel-cage. 

The  curves  shown  in  Fig.  100  may  be  considered  as  an  example 
of  the  extreme  opposite  condition.     This  represents  the  currents  in 


THE  SINGLE-PHASE  INDUCTION  MOTOR  199 

the  stator  and  rotor  of  a  three-phase  motor,  wound  with  full  pitch  coils, 
and  operated  on  a  single-phase  circuit.  The  curve  of  distribution  of 
the  stator  current  is  of  course  rectangular  as  shown.  The  resultant 
band  of  current  is  sinusoidal,  and  the  rotor  current  is  of  the  proper 
value  to  give,  in  combination  with  the  stator  current,  the  resultant 
harmonic  band  of  current.  As  before,  it  will  be  seen  that  the  rotor 
current  sheet  moves  in  the  opposite  direction  from  the  resultant  current 
sheet,  but  that  it  is  now  very  much  distorted  from  the  sine  shape. 

These  curves,  like  those  of  Fig.  99,  are  for  the  no-load  condition. 
With  the  motor  under  load,  the  value  of  the  rectangular  stator  current 
would  be  increased  in  proportion  to  the  current.  There  would  be 
added  in  the  rotor  a  corresponding  rectangular  current  distribution, 
almost  exactly  equal  and  opposite  to  the  added  stator  current.  In 
the  case  of  Fig.  99  a  corresponding  sine  distribution  of  current  would 
be  added.  It  is  this  added  component,  in  connection  with  the  com- 
ponent of  the  flux  at  right  angles  to  the  direction  of  the  stator  winding, 
that  produces  the  rotor  torque. 

Returning  to  the  ideal  case  of  sinusoidal  distribution,  as  shown  in  Fig. 
99,  instead  of  considering  the  rotor  current  sheet  as  a  band  of  current 
rotating  backward  in  space,  we  may  perhaps  gain  a  better  idea  of  the 
phenomena  involved  if  we  separate  the  band  of  rotor  current  into 
two  component  current  sheets,  each  stationary  in  space  but  varying 
harmonically  in  magnitude.  The  bands  differ  90  degrees  in  time  phase 
and  are  displaced  90  degrees  in  space.  As  we  have  already  shown,  the 
combination  of  two  such  stationary  bands  is  equivalent  to  one  rotating 
band.  Each  band  of  current  may  be  represented  by  a  vector  as  shown 
in  Fig.  10 1.  The  resultant  current  sheet  will  be  of  constant  value,  and 
will  rotate  as  shown. 

Under  the  condition  of  no-load  and 
synchronous  speed,  we  have  just  seen  that 
we  have  in  the  stator  a  current  sheet  of 
double  the  value  of  the  rotor  current  sheet. 
This  sheet  can  be  represented  by  a  vector 
of  double  the  value  of  one  of  the  vectors 
representing  the  rotor  current  sheets  and 
at  an  angle  of  180  degrees  with  one  of  them. 

The  relations  are  then  as  shown  in  Fig.  102,      ^  IOI>_VectorDiagiam 
and  the  direction  of  rotation  of  the  resultant      of  Current  Bandg  jn  Rotor 
of  the  three  current  sheets    will   be    in    the 
opposite  direction  to  that  of  the  rotor  current  sheet.      The  net   result 


200 


THE  INDUCTION  MOTOR 


Stator y 


.  Resultaut 


then  is  that  we  may  consider  that  we  have  three  stationary  current 
sheets,  one  in  the  stator  and  two  in  the  rotor.  One  of  the  rotor  sheets 
is  directly  opposed  to  and  offsets  half  of 
the  stator  current  sheet.  This  resultant 
then  combines  with  the  remaining  rotor 
sheet  to  form  a  rotating  current  sheet  of 
constant  value.  This  rotating  current 
sheet  sets  up  a  corresponding  rotating 
flux  sheet  which  is  likewise  constant  in 
value  and  rotates  in  synchronism  with 
the  current  sheet.  The  above  applies  of 
course  to  the  no-load  condition  only. 

Fig.  103  represents  a  type  of  single- 
phase  motor  to  which  the  preceding 
analysis  particularly  applies.  The  coil 
marked  P  represents  the  stator  winding. 
The  motor  represented  diagrammatically 
is  of  the  two-pole  variety,  and  of  course 
in  the  actual  machine  half  of  the  stator 
winding  would  be  on  each  side  of  the 
stator  core.  The  rotor  is  of  the  same 
construction  as  the  armature  of  a  direct- 
current  machine.  It  has  a  commutator 
and  as  shown  has  in  a  two-pole  machine 
four  brushes  arranged  90  degrees  apart  and  short-circuited  across  both 
diameters.  If  the  motor  has  more  than  two  poles,  the  number  of 
brushes  required  would  be  twice  the  number  needed  with  the  same 
armature  and  commutator  used  as  a  direct-current  armature.  For 
the  sake  of  simplicity,  we  may  regard  the  winding  as  being  a  ring 
winding.  This  is  convenient  since  in  this  case  the  m.m.f.  of  either 
circuit  is  in  line  with  the  corresponding  brushes,  or  in  other  words,  if 
there  is  a  current  through  either  set  of  brushes  the  corresponding  poles 
of  the  armature  will  be  in  line  with  the  brushes  used. 

An  armatue  of  this  sort,  if  used  in  a  polyphase  field,  will  operate 
in  much  the  same  manner  as  would  a  squirrel-cage  rotor.  We  have 
studied  this  action  somewhat  in  Chapter  VI.  If  used  in  a  single-phase 
stator,  it  will  likewise  operate  much  as  would  a  squirrel-cage  rotor. 
The  current  forming  the  current  sheet  T  will  be  through  the  brusher 
T  and  T',  that  forming  the  sheet  S  through  .S  and  Sf. 

There  is  another  way  in  which  we  may  consider  the  action  of  the 


FIG.  102. — Vector  Diagram  of 

Current  Bands  in  Rotor  and 

Stator. 


THE  SINGLE-PHASE  INDUCTION   MOTOR 


201 


single-phase  motor,  and  which  is  very  convenient  in  developing  the 
formulae  of  the  motor.  This  method  consists  in  brief  in  considering  the 
flux  as  being  resolved  into  components  and  considering  separately 
the  effect  of  each  of  these  components.  Thus  in  Fig.  103,  we  may 
assume  that  we  have  a  flux  in  the  direction  TT'  due  to  the  resultant 
of  the  current  through  the  stator  coil  and  the  rotor  current  in  TT'. 
There  is  also  a  flux  in  the  direction  SS'  due  to  the  current  through  the 
brushes  SS'.  These  two  fluxes  are  of  equal  value  but  differ  90  degrees 
both  in  phase  and  in  space. 
Their  resultant  is  therefore  a 
rotating  magnetic  field.  The 
current  in  SS'  is  called  the 
speed  current  and  the  cor- 
responding field  the  speed 
field.  In  all  the  conductors 
each  way  from  5  to  S'  there 
will  be  generated  an  e.m.f. 
proportional  to  the  speed  and 
to  the  value  of  the  flux  in 
the  direction  TT'  at  the 
instant  under  consideration. 
This  e.m.f.  will  then  be  in 
phase  with  the  flux  TT'.  The 
wave  will  be  sinusoidal  if 
the  flux  variation  along  TT' 
is  sinusoidal.  This  condition 
is  usually  assumed  in  theoret- 
ical discussions. 

The  resultant  current  along 
SS'  will  lag  nearly  90  de- 
grees behind  the  generated  e.m.f.  since  the  rotor  is  highly  inductive 
in  this  direction.  The  flux  in  the  direction  SS'  will,  except  for  the 
slight  effective  hysteresis,  be  in  phase  with  the  current.  It  is  therefore 
90  degrees  behind  the  e.m.f.  in  SS'  and  likewise  90  degrees  behind  the 
flux  in  TT'. 

The  flux  TT'  is  of  nearly  constant  value,  irrespective  of  the  speed 
of  rotation  of  the  rotor.  That  this  is  so  is  apparent  when  we  consider 
that  the  counter  e.m.f.  in  the  stator  coil  is  nearly  equal  to  the  applied 
e.m.f.  and  hence  the  flux  producing  this  e.m.f.  must  also  be  nearly 
constant  to  produce  a  constant  back  e.m.f.  The  difference  between 


FIG.  103. — Single-phase   Commutator  Type 
Induction   Motor. 


202  THE  INDUCTION  MOTOR 

the  applied  and  counter  e.m.f.  is  due  to  the  resistance  drop  in  the  stator 
and  the  drop  due  to  the  local  leakage  reactance,  i.e.,  that  due  to  the 
lines  which  cut  the  primary  coil  out  do  not  pass  through  the  rotor. 
Notwithstanding  these  two  drops,  the  counter  e.m.f.  and  consequently 
the  flux  in  the  circuit  TT'  are  nearly  constant  during  normal  operation. 

Regarding  the  flux  SS'  the  case  is  not  quite  so  simple.  In  cutting 
the  flux  TT'  a  certain  e.m.f.  will  be  generated  along  the  axis  SS'. 
There  will  be  sufficient  current  to  produce  a  flux  which  will  generate 
by  transformer  action  approximately  the  same  e.m.f.  The  flux  in 
SS'  will  therefore  be  approximately  proportional  to  the  e.m.f.  generated 
in  SS'  by  the  cutting  of  the  flux  TT'.  If  the  motor  is  operating  at 
synchronous  speed  the  same  e.m.f.  will  be  generated  whether  it  is  due 
to  the  passage  of  the  conductors  through  the  flux  or  due  to  the  change 
of  the  flux  through  the  circuit  of  the  conductors  as  in  a  transformer. 
In  other  words,  it  makes  no  difference  whether  the  conductors  are  at 
rest  and  the  flux  in  motion,  or  the  flux  is  at  rest  and  the  conductors  in 
motion.  Hence  at  synchronous  speed  the  transformer  field  TT'  and 
the  speed  field  SS'  are  approximately  equal.  At  any  other  speed  the 
e.m.f.  generated  by  the  motion  of  the  conductors  will  be  reduced  or 
increased  in  proportion  to  the  speed,  and  consequently  the  field  SS' 
will  be  changed  in  the  same  proportion.  If  we  let  5  equal  the  speed 
in  per  cent  of  synchronism  and  designate  the  two  fields  by  T  and  5" 
we  have  without  serious  error,  S  =  sT. 

In  a  similar  way  we  have  in  the  circuit  TT'  two  e.m.fs.,  one  due 
to  the  transformer  action  of  the  flux  and  the  other  to  the  cutting  of  the 
conductors  through  the  flux  SS'.  At  synchronism,  these  are  as  before 
approximately  equal  and  opposite,  the  difference  being  just  enough  to 
establish  the  necessary  current. 

When  load  is  applied  to  the  motor,  the  rotor  slows  down  somewhat. 
The  speed  field  S  is  reduced  to  S=sT.  The  back  e.m.f.  in  circuit 
TT'  due  to  the  cutting  of  the  flux  in  the  circuit  SS'  is  then  reduced  to 
s2T  and  this  establishes  more  current  through  the  circuit  TT'.  The 
current  in  SS'  is  slightly  reduced.  The  increased  current  along  TT' 
has  a  demagnetizing  effect  in  the  direction  TT'  and  this  immediately 
allows  more  current  to  flow  through  the  stator  coil  to  offset  its  action, 
and  the  flux  remains  at  nearly  its  former  value. 

The  current  in  the  circuit  TT'  can  exert  no  torque  with  the  flux 
TT',  and  the  same  is  true  of  the  current  and  flux  in  the  circuits  SS'. 
Conversely,  current  in  either  of  the  two  circuits  is  in  the  proper  mechan- 
ical position  to  produce  torque  with  the  field  corresponding  to  the 


THE  SINGLE-PHASE  INDUCTION    MOTOR 


203 


other  circuit.  The  truth  of  these  statements  will  readily  appear  from 
a  consideration  of  Fig.  104  and  105.  In  these  drawings  the  motor  is 
represented  as  though  it  were  provided  with  four  projecting  poles. 
The  student  must  keep  in  mind  that  in  the  actual  motor,  the  air  gap  is 
equal  all  around  the  rotor,  and  that  the  motor  as  illustrated  is  of  the 
two-pole  type. 

With  a  current  through  the  brushes  SS'  the  distribution  of  current 
in  the  rotor  would  be  as  shown  in  Fig.  104.  The  circles  represent 
currents  directed  toward  the  observer,  and  the  crosses  currents  from 
him.  It  will  be  seen  that  the  speed  current  and  the  transformer  field 
will  react  to  produce  torque,  and  the  same  is  true  of  the  transformer 


Transformer  Field 


Transformer  Field 


FIG.  104. — Relation  of  Current  and 
Flux  in  Single-phase  Motor. 


FIG.  105. — Relation  of  Current  and 
Flux  in  Single-phase  Motor. 


current  and  the  speed  field.     This  is  so  since  all  the  conductors  lying 
in  a  field  of  given  polarity  carry  currents  in  the  same  direction. 

In  addition  to  this  fact,  however,  we  must  consider  the  effect  of 
phase  displacement.  While  the  currents  and  fluxes,  as  shown  in  Fig. 
104,  are  in  the  best  mechanical  position  to  produce  torque,  the  actual 
average  torque  produced  may  be  in  either  direction  or  may  even  be 
zero.  This  latter  is  in  fact  nearly  the  case  with  the  speed  current 
and  the  transformer  field.  The  speed  current  is  (except  for  the  effect 
of  hysteresis)  in  phase  with  the  speed  flux.  This  latter,  as  we  have 
pointed  out,  differs  90  degrees  in  time -phase  from  the  transformer  flux, 
or  the  speed  current  is  90  degrees  in  time-phase  from  the  transformer 
field.  Hence  at  the  time  the  current  is  a  maximum  the  field  is  zero  and 


204  THE  INDUCTION    MOTOR 

vice  versa.  Moreover,  if  we  consider  the  moment  when  the  flux  is  a 
maximum  and  the  current  is  zero,  it  will  be  seen  that  during  the  previous 
quarter  revolution  the  torque  has  been  in  one  direction  and  that  it 
will  be  in  the  opposite  direction  during  the  succeeding  quarter  revolu- 
tion, since  the  current  will  be  in  the  reverse  direction.  Hence  it  will 
be  readily  apparent  that  the  average  torque  will  be  zero  as  far  as  the 
speed  current  and  the  transformer  field  are  concerned. 

Considering  now  Fig.  105,  the  conditions  at  no-load  are  nearly 
the  same  as  those  in  the  case  of  Fig.  104.  The  current  in  the  stator 
coil  being  a  magnetizing  current  is  nearly  in  phase  with  the  transformer 
flux.  The  rotor  transformer  current  is  nearly  opposite  in  phase  to  the 
stator  current.  Hence  it  is  90  degrees  out  of  phase  with  the  speed  flux, 
and  as  explained  can  exert  no  torque  with  it.  This  was  of  course 
to  be  expected,  since  at  no-load  the  only  torque  required  is  enough  to 
overcome  the  losses  of  the  machine. 

As  the  load  on  the  motor  is  increased  the  current  in  the  stator 
winding  is  increased  by  the  addition  of  a  component  of  current  in  phase 
with  the  applied  e.m.f.  This  added  current  is  consequently  nearly 
90  degrees  different  in  phase  from  the  transformer  flux,  or  in  phase 
with  the  speed  flux.  There  will  be  produced  in  the  rotor  a  current 
of  the  same  m.m.f.  as  the  added  primary  current  and  this  will  be  in 
opposition  to  the  primary  current,  and  hence  it  will  likewise  be  in  the 
best  phase  to  produce  torque  with  the  speed  field.  The  total  torque  of 
the  motor  is  then  due  to  this  added  rotor  current  and  to  the  speed  field. 

If  we  represent  the  torque  by  D,  and  the  added  rotor  current  by 
Ia  we  have,  assuming  suitable  units, 

secondary  input  =  PI  =  IaT. 

In  the  same  units,  the  torque  in  synchronous  watts  is  given  by  the  equa- 
tion, 

Torque  =  D  =  IaS=sTIa. 
Also, 

output  =  P2  =  sD  =  sTIa. 
Solving  for  5  we  get, 

P2      s2TIa  If£ 

PTT77=5'    °r    5=\fe~; 

or,  the  speed  in  per  cent  of  synchronism  is  equal  to  the  square  root  of 
the  secondary  efficiency. 


THE  SINGLE-PHASE  INDUCTION    MOTOR  205 

The  slip  is  given  by  i  —  s,  and  it  will  be  readily  seen  that  if  the  speed 
is  near  synchronism,  the  slip  in  percentage  of  the  synchronous  speed 
is  equal  to  twice  the  percentage  of  rotor  loss.  Thus  if  the  speed  is 
97  per  cent,  we  have: 

P7  PI 

°'97  °r 


hence,  we  see  that  the  slip  is  3  per  cent  and  the  secondary  loss  is  5.91 
per  cent  or  nearly  double  the  slip. 

It  must  be  kept  in  mind  that  the  loss  referred  to  is  that  due  to  the 
added  component  of  the  rotor  current  and  is  in  fact  a  part  only  of  the 
loss  in  the  circuit  TT'.  In  addition  to  this,  there  is  a  loss  in  both  TT' 
and  SS'  due  to  the  magnetizing  components  of  the  currents.  The  error 
in  our  conclusion  is  particularly  apparent  at  synchronous  speed  when, 
according  to  this  conclusion,  we  should  have  no  rotor  loss,  while  in 
reality  there  is  a  considerable  loss  due  to  the  magnetizing  currents.  At 
a  reasonably  large  load,  the  error  becomes  slight. 

It  will  be  remembered  that  in  the  case  of  the  polyphase  motor  the 
slip  is  equal  to  the  rotor  loss.  Thus  in  the  single-phase  motor,  the 
rotor  loss  is  approximately  double  that  of  the  polyphase  motor  for  the 
same  slip. 

There  is  another  method  of  considering  the  single-phase  induction 
mo  tor,  which,  while  apparently  somewhat  artificial,  leads  in  some  cases  to 
very  simple  deductions.  This  method  is  based  on 
the  proposition  that  an  alternating  field,  stationary 
in  space  ,  may  be  considered  as  being  made  up  of 
two  fields  each  of  half  the  value  of  the  stationary 
field,  rotating  in  space  in  opposite  directions  with 
equal  velocity.  Thus  in  Fig.  106,  if  <f)  represents 
the  instantaneous  value  of  the  alternating  field, 
we  may  consider  it  as  being  made  up  of  the  two 
fields  (j)a  and  fa  rotating  as  shown.  The  vector 
sum  of  the  two  fields  will  at  all  times  be  equal  to 

the  value   of  the   stationary  field.     Thus  we  may      FlG'  Io6--R^olu- 

tion  of  a  Stationary 

think  of  the   rotor  of  the   single-phase   motor  as      Flux  into  Two  Re- 
being  at  the  same  time  under  the  influence  of  two         volving  Fluxes. 
magnetic    fields,    rotating   in   opposite  directions. 
The  two  fields  are  of  equal  strength  and  half  the   value  of  the  actual 
stationary  field. 

With  this  in  mind,  we  can  very  readily  deduce  the  speed  torque 


206 


THE  INDUCTION   MOTOR 


curves  of  the  single -phase  motor.  Thus  in  Fig.  107  are  drawn  several 
speed-torque  curves  of  a  wound-rotor  polyphase  induction  motor.  The 
curve  A  is  that  of  the  motor  when  no  external  resistors  are  connected 
in  the  rotor  circuit.  The  curves  B  and  C  are  for  larger  values  of  the 
external  resistance.  The  derivation  of  these  curves  was  fully  explained 
on  page  73.  The  curves  are  extended  below  the  zero  line  to  indicate 
the  torque  encountered  in  case  the  rotor  is  forced  to  rotate  in  the 
opposite  direction  to  that  in  which  it  tends  to  turn.  The  curves  A', 
B'  and  C'  indicate  the  speed-torque  curves  of  the  same  motor  with  the 
same  resistors  in  the  rotor  circuit  when  it  is  operated  in  the  opposite 
direction. 

Suppose  now  that  we  have  a  single-phase  motor  operating  at  the 
speed  represented  by  OK.  From  what  has  been  said,  it  is  evident 
that  we  may  consider  that  the  motor  is  subjected  to  the  influence  of 
two  fields  rotating  in  opposite  directions.  The  one  field  will  exert 
the  torque  KM,  while  the  other  field  will  oppose  this  torque  with  the 
torque  KL.  The  net  torque  is  the  resultant  of  the  two  or  LM.  In  Fig. 
108  the  length  LM  representing  the  torque  at  the  speed  OK,  is  laid  off 
as  shown.  In  the  same  way  any  number  of  points  may  be  obtained. 
As  was  to  be  expected,  the  torque  at  standstill  is  zero. 

In  a  similar  manner,  we  can 
derive  the  curves  for  a  single- 
phase  motor  with  various  values 
of  resistance  in  the  rotor  circuit. 
Thus  by  combining  the  curves 
B  and  B'  and  C  and  C'  of  Fig. 
107,  we  obtain  the  curves  B  and 
C  of  Fig.  108. 

On  examining  the  set  of 
curves  of  Fig.  108  several  new 
facts  are  at  once  apparent: 

i.  With  a  given  magnetic 
field  and  a  given  rotor,  the 
rated  output  can  not  be  greater 
than  half  that  of  the  same  motor 
operated  polyphase.  This  is  ap- 
parent since  if  we  are  limited  to 
a  certain  rotor  loss  the  torque 
will  be  less  than  half  as  great, 


K 

H-50* 


-50* 


-100* 


FIG.  107. — Speed  Torque  Curves  of 
Polyphase  Induction  Motor. 


since  each  of  the  two  component  fields  is  of  half  the  strength    of    the 


THE  SINGLE-PHASE  INDUCTION    MOTOR 


207 


corresponding  polyphase  field,  and  since  there  is  a  certain  back  torque 
due  to  the  field  which  is  rotating  in  the  reverse  direction. 

2.  The  torque,  and  to  a  still  greater  extent  the  output,  will  be 
greater  the  less  the  secondary  resistance.     These  facts  are  apparent 
from    the    curves.     In    this    re- 
spect,   the    single-phase     motor 

differs  from  the  polyphase  motor 
in  which  the  maximum  torque 
is  independent  of  the  secondary 
resistance. 

3.  The    plain     single-phase 
induction  motor  has  no  starting 
torque.    If,  however,  it  be  given 


FIG.  1 08. — Speed  Torque  of  Single- 
phase  Induction  Motor. 


is  not  too  great,   the   motor    will 
motors    are  sometimes    started  in 


a   start    in   either   direction,    a 

small  torque  will    be  developed 

in  that  direction  and    if  the  load 

accelerate    to    full    speed.     Small 

this  way,  but  this  procedure  is  very  unsatisfactory  except  in  the  case  of 

motors  of  very  small  size,  and  for  use  in  applications  where  the  required 

starting  torque  is  very  small. 

4.  Although  the  maximum  torque  is  reduced  by  increasing  the 
resistance  of  the  secondary  by  inserting  resistors,  the  speed  at  which 
this  torque  is  developed  is  also  reduced.  Hence  the  use  of  a  moderate 
amount  of  resistance  in  the  rotor  will  help  the  starting  conditions, 
although  the  gain  is  by  no  means  so  great  as  in  the  case  of  the  polyphase 
motor.  In  American  practice,  external  resistors  in  the  rotor  circuit  to 
assist  in  starting  the  motor  are  rarely  or  never  used.  Abroad,  a  number 
of  motors  have  been  built  in  which  use  is  made  of  such  an  expedient. 


CHAPTER  XIV 

THE   SINGLE-PHASE   INDUCTION   REPULSION   MOTOR 

THE  line  of  demarkation  between  the  plain  single -phase  induction 
motor,  as  previously  described,  and  the  various  types  of  motors  usually 
described  under  the  name  of  commutator  type  single-phase  motors, 
is  so  slight  that  one  is  constantly  tempted  to  overstep  the  line.  In 
general,  the  commutator  type  single-phase  motors  have  been  developed 
with  the  idea  in  mind  of  using  the  motors  in  traction  work.  They 
have  therefore  had  "series"  characteristics,  that  is,  if  the  load  is  removed 
the  speed  will  increase  almost  without  limit,  and  on  the  other  hand, 
If  the  load  be  increased,  the  motor  slows  down  very  greatly  and  at  the 
same  time  greatly  increases  its  torque.  The  plain  single -phase  induc- 
tion motor  on  the  other  hand,  has  a  speed-torque  characteristic  of 
such  a  type  that,  as  the  load  is  increased,  the  speed  decreases  only 
to  a  slight  extent.  A  motor  the  speed  of  which  varies  in  this  way  is 
said  to  have  a  shunt  characteristic.  A  motor  of  this  type  is  desired  for 
the  greater  number  of  applications,  aside  from  railroad  work  and  crane 
work. 

The  plain  squirrel-cage  single -phase  induction  motor  has  shunt 
characteristics  and  is  therefore  available  for  general  purposes.  It 
suffers,  however,  in  comparison  with  the  shunt-wound  direct-current 
motor  in  having  a  power-factor  less  than  one ;  in  having,  in  its  simplest 
form,  no  starting  torque,  and  in  having  only  a  small  starting  torque 
even  with  special  starting  appliances,  and  in  not  being  capable  of  having 
its  speed  adjusted  to  different  values.  Recently,  considerable  activity 
has  been  apparent  among  inventors  in  an  attempt  to  overcome  these 
deficiencies.  These  attempts  in  general  involve  the  use  of  a  commutator 
with  its  obvious  disadvantages. 

Considering  the  motor  shown  in  Fig.  103,  it  is  apparent  that  it 
has  approximately  the  same  characteristics  as  the  plain  squirrel-cage 
single-phase  induction  motor.  In  fact,  we  have  seen  that  all  the  char- 
acteristics of  the  squirrel-cage  machine  can  be  derived  from  a  con- 
sideration of  this  form  of  motor.  It  likewise  has  no  starting  torque, 

208 


THE  SINGLE-PHASE  INDUCTION  REPULSION  MOTOR         209 


and  since  it  has  no  advantage  and  suffers  from  the  presence  of  the  com- 
mutator, it  is  clearly  not  adapted  to  general  use.  By  a  slight  shift  of 
the  brushes  it  is  however  possible  to  give  the  motor  a  good  starting 
torque. 

To  understand  this,  consider  what  would  happen  in  the  motor  of 
Fig.  103,  if  of  the  two  circuits  between  the  two  sets  of  brushes  only  one, 
say  TT',  were  closed.  This  circuit  is  in  the  best  position  relative  to 
the  stator  winding  to  have  current  produced  in  it,  and  consequently 
if  the  circuit  TT'  alone  were  closed  and  current  were  applied  to  the 
motor  at  standstill,  there  would  be  a  large  current  established  in  the 
rotor  circuit.  It  is,  however, 
apparent  that  the  current  in 
the  rotor  would  be  so  related 
to  the  primary  flux  that  the 
resultant  torque  would  be 
zero.  If,  on  the  other  hand, 
only  the  circuit  SS'  were 
closed,  no  current  would  be 
produced  in  it  at  standstill, 
and  consequently  there  would 
be  no  torque.  It  is  however 
true,  that  if  there  were  cur- 
rent in  SS'  it  would  be  in 
the  best  possible  position  to 
produce  torque  in  connection 
with  the  primary  flux. 

If  now  one  of  the  circuits 
between  the  two  brushes  be 
closed  and  the  brushes  be 
set  in  any  position  except  in 
line  with  or  at  right  angles 
to  the  line  of  the  stator  coil,  as  shown  in  Fig.  109,  there  will  be  a  resultant 
torque.  A  motor  of  this  type  with  only  the  one  set  of  brushes  on  the 
commutator  is  known  as  a  repulsion  motor.  Such  motors  are  used  to 
some  extent  in  railroad  work.  They  have,  however,  series  character- 
istics, that  is  the  speed  increases  almost  without  limit  as  the  load  is 
decreased  and  hence  are  not  suited  to  the  class  of  service  we  are  at  present 
considering.  This  principle  is,  however,  utilized  in  a  number  of  success- 
ful single-phase  induction  motors  as  a  means  of  starting.  In  this  case, 
some  form  of  device,  usually  a  couple  of  rotating  weights  operating  by 


FIG.  109. — Connections  of  Repulsion 
Motor. 


210 


THE   INDUCTION   MOTOR 


centrifugal  action,  is  used  to  remove  the  brushes  from  the  commutator 
at  the  proper  speed,  and  at  the  same  time,  short-circuit  all  the  segments 
of  the  commutator.  In  this  way,  the  machine  is  transformed  into  an 
ordinary  single -phase  induction  motor  with  a  rotor  which  is  practically 
equivalent  to  a  squirrel-cage  rotor. 

Returning  now  to  the  type  of  motor  represented  in  Fig.  103,  but 
giving  the  brushes  a  small  lead  in  the  one  direction  or  the  other,  as  shown 
in  Fig.  no,  it  will  be  apparent  that  with  either  set  of  brushes  alone, 

there  would  be  a  starting 
torque.  With  both  sets  of 
brushes  in  use,  there  will  be 
a  resultant  torque,  due  to  the 
combined  action  of  the  two. 
With  the  angle  as  shown, 
the  action  of  the  brushes 
TT  will  be  far  greater  than 
that  of  SSr,  and  the  direction 
of  rotation  would  be  clock- 
wise. If  the  shift  of  the 
brushes  had  been  in  the  op- 
posite direction,  the  direction 
of  rotation  would  also  have 
been  reversed. 

This  motor,  unlike  the 
repulsion  motor,  does  not 
tend  to  increase  in  speed  in- 
definitely but  approaches  a 
limiting  speed  somewhat  near 
to  the  synchronous  speed.  As 
we  have  seen,  if  the  brushes 
are  in  line  with  the  stator 

coil,  the  machine  will  act  in  all  respects  like  an  induction  motor  and 
will  have  a  maximum  speed  somewhat  below  synchronism.  This  is  not 
the  case  with  the  motor  represented  in  Fig.  no.  The  no-load  speed 
of  the  motor  will  be  somewhat  above  synchronous  speed,  while  the 
full-load  speed  will  in  general  be  below  synchronism. 

So  far,  we  have  arrived  at  a  single -phase  induction  motor  having 
shunt  characteristics,  and  possessing  a  fairly  large  starting  torque. 
With  this  construction  it  is,  however,  possible  at  a  slight  additional 
expense,  to  improve  the  power-factor  of  the  motor.  This  is  done  by 


FIG.  no. — Single-Phase  Induction  Motor 
with  Brushes  Displaced  to  give 
Starting  Torque. 


THE  SINGLE-PHASE  INDUCTION  REPULSION  MOTOR        211 

adding  an  additional  coil  wound  in  the  same  slots  as  the  main  winding 
and  connected  to  the  brushes  SS'.  The  connections  are  shown  in  Fig. 
in. 

Instead  of  using  a  separate  compensating  coil,  it  is  evident  that  a  por- 
tion of  the  main  stator  winding  may  be  used  instead.  This  connection 
is  shown  in  Fig.  112. 

A  comparison  of  this  connection  with  that  of  the  polyphase  motor 
shown  in  Fig.  49,  will  show  that  the  connections  are  essentially  the  same. 


FIG  in. — Single-phase.  Compensated  FIG.  112. — Single-phase  Compensated 

Induction  Motor.  Induction  Motor. 

The  action  of  the  compensating  coil  and  brushes  C  in  improving 
the  power-factor  is  practically  the  same  as  the  action  of  the  auxiliary 
brushes  in  the  case  of  the  polyphase  motor.  It  will  be  remembered  that 
there  are  in  the  circuit  SS'  two  approximately  equal  and  opposite 
e.m.fs.,  the  transformer  e.m.f.  and  the  speed  e.m.f.,  both  of  these  are  in 
phase  with  the  flux  in  the  circuit  TT'  and  consequently  in  quadrature 
with  the  primary  e.m.f.  The  current  in  the  circuit  SS'  is  in  quadra- 
ture with  both  of  these  e.m.fs.,  and  of-  course  in  phase  with  the  flux  in 
the  same  circuit.  The  e.m.f.  applied  from  the  compensating  coil  is 
therefore  approximately  in  phase  with  the  current  and  flux,  and  at 
right  angles  to  the  two  e.m.fs.  already  present  in  the  SS'  circuit. 


212  THE  INDUCTION  MOTOR 

We  can  perhaps  best  appreciate  the  effect  of  this  added  e.m.f. 
by  considering  that  the  action  as  already  described  goes  on  as  before, 
but  that  since  we  have  now  added  a  new  e.m.f.  in  the  speed  circuit, 
there  will  be  produced  a  new  flux  so  as  to  generate  by  transformer 
action  a  new  e.m.f.  approximately  equal  and  opposite  to  that  which  we 
have  introduced  from  the  compensating  coil.  This  new  flux  is  at  right 
angles  in  time-phase  to  the  flux  formerly  present,  but  in  the  same 
position  as  regards  the  core.  We  may  call  this  added  flux  the  compen- 
sating flux. 

The  compensating  flux,  being  at  right  angles  to  the  speed  flux,  has 
no  effect  on  the  torque  of  the  motor,  and  hence  has  little  or  no  effect 
on  the  speed.  It  has,  however,  the  effect  of  introducing  a  new  speed 
e.m.f.  in  the  transformer  axis  TT'.  This  new  e.m.f.,  which  we  may 
call  by  analogy  the  compensating  e.m.f.,  being  at  right  angles  to  the 
speed  and  transformer  e.m.f  .s  in  the  transformer  axis,  tends  to  combine 
with  their  resultant  to  form  a  new  resultant  e.m.f.  We  may,  however, 
as  before  consider  the  effect  of  this  e.m.f.  by  itself.  Both  the  speed 
and  the  transformer  e.m.f.  in  the  TT'  circuit  are  in  phase  with  and  in 
phase  opposition  to  the  primary  counter  generated  e.m.f.  and  nearly 
so  with  the  primary  applied  e.m.f.  The  compensating  e.m.f.  is  there- 
fore nearly  at  right  angles  to  the  primary  applied  e.m.f.  that  was  present 
before  its  introduction.  The  compensating  e.m.f.,  however,  tends  to 
set  up  a  current  and  a  flux  nearly  at  right  angles  to  itself,  the  flux  being 
of  such  a  value  as  to  generate  by  transformer  action,  approximately  the 
same  e.m.f.  as  the  compensating  e.m.f.  This  flux  in  turn  generates 
a  new  e.m.f.  in  the  stator  approximately  at  right  angles  to  the  transformer 
voltage  which  would  be  present  without  it.  The  total  counter  e.m.f. 
of  the  primary  circuit  is  the  resultant  of  these  two  e.m.fs.  and  con- 
sequently its  phase  can  be  changed  so  as  to  lead  the  primary  current 
by  a  greater  angle,  or  if  is  applied  in  the  opposite  direction,  the  e.m.f. 
may  be  made  to  lag  behind  the  primary  current  or  to  be  in  phase  with 
it;  that  is,  the  motor  may  be  made  to  take  a  current  in  phase  with  the 
applied  e.m.f.  or  even  a  leading  current. 

In  Fig.  113  are  shown  the  characteristic  curves  oi  a  small  General 
Electric  motor  of  this  type.  The  connections  are  as  shown  in  Fig. 
112.  It  will  be  seen  that  the  power-factor  is  high  for  all  loads.  At 
no-load,  the  compensating  brushes  and  the  main  brushes  each  carry 
current.  These  two  currents  are  in  quadrature  and  serve  to  furnish  the 
magnetizing  currents  in  the  two  axes  of  the  motor.  The  stator  current 
is  comparatively  small,  and  serves  principally  to  furnish  by  transforma- 


THE  SINGLE-PHASE  INDUCTION  REPULSION  MOTOR        213 

tion  the  current  required  for  the  compensating  brushes.     As  the  load 
is  increased,  the  current  in  the  primary  coil  is  increased  by  a  component 


100 


H.P.  putput 


1600 


too 


.5  1.  1.5  2.  2.5  3. 

FIG.  113. — Characteristic  Curves  of  General  Electric  Type  R.I.  Motor. 

in  phase  with  the  e.m.f.,  and  the  current  in  the  main  brushes  is  corres- 
pondingly increased. 


214 


THE  INDUCTION  MOTOR 


The  current  in  the  compensating  brushes  on  the  other  hand  con- 
stantly decreases,  reaches  nearly  zero  and  finally  increases.  At  the 
point  of  smallest  current  through  the  compensating  winding,  the  brushes 
SS'  might  be  removed,  and  the  conditions  of  running  would  obviously 
not  be  changed.  In  other  words,  the  motor  at  this  particular  load  is 
operating  practically  as  a  repulsion  motor.  At  all  other  loads,  it  is 
forced  to  operate  at  either  a  higher  or  a  lower  speed  than  would  be  the 
case  as  a  repulsion  motor. 


SPEED-CONTROL  OF  SINGLE-PHASE  MOTORS 

A  single-phase  motor  of  this  type  admits  of  efficient  speed  control 
through  at  least  a  limited  range  of  speed.  The  principles  upon  which 
this  is  based  are  similar  to  those  explained  in  connection  with  the  poly- 
phase motor.  The  single-phase  motor  however  admits  of  some  exten- 
sion of  these  methods,  and  will  therefore  be  described  separately. 

In  the  following,  the  diagrams  of  the  motors  will  be  drawn  without 
the  compensating  coil  for  raising  the  power-factor.  It  will,  however, 
be  understood  that  in  practice  such  a  coil  would  usually  be  provided 
in  addition  to  the  windings  as  shown. 

There  are  in  general  two  methods  of  speed  control,  and  these  may 
be  called  by  analogy  with  the  shunt-wound  direct-current  motor,  the 

armature    control    and   field    control. 

To  change  the  speed  by  the  first 
method,  an  e.m.f.  of  the  primary 
frequency  and  of  the  same  phase  as 
the  applied  e.m.f.  must  be  introduced 
into  the  TT'  circuit.  To  employ  the 
method  of  field  control  an  e.m.f. 
differing  in  phase  90  degrees  from  the 
primary  e.m.f.  must  be  introduced  into 
the  SS'  circuit. 

In  Figs.  114  and  115  are  shown 
two  examples  of  speed  control  by 
the  first  method.  In  Fig.  114  taps  are 
brought  out  from  the  primary  winding 
and  the  e.m.f.  between  these  points  is 
introduced  into  the  TT'  circuit.  In 
Fig.  115  an  auto-transformer  is  pro- 
vided to  give  the  required  voltage.  Instead  of  an  auto-transformer 


FIG.  114. — Single-phase  Motor  with 
Armature  Speed  Control. 


THE  SINGLE-PHASE  INDUCTION  REPULSION  MOTOR        215 

an  ordinary  transformer  with  primary  and  secondary  coils  might 
have  been  provided.  It  will  be  remembered  that  we  have  in  the 
TT'  circuit  two  voltages,  one  due  to  the  transformer  action  of  the 
primary  flux,  the  other  due  to  the  cutting  of  the  flux  of  the  speed 
field.  With  the  brushes  TT'  short-circuited  these  two  are  ap- 
proximately equal  and  opposite  and  at  right  angles  to  the  primary 
flux.  They  are  consequently  in  phase  and  phase  opposition  respect- 
ively to  the  primary  applied  e.m.f.  If  we  add  another  e.m.f. 
in  phase  with  the  primary  voltage,  we  destroy  this  equality  and  there 
will  be  a  large  current  until  a  balance  is  re-established.  This  current 


FlG.  115. — Single-phase  Motor  with  Armature  Speed  Control. 

will  be  nearly  in  phase  with  the  primary  e.m.f.  The  flux  in  the  SS' 
axis  is  in  the  same  phase  and  consequently  the  current  in  the  TT'  axis 
is  in  the  best  position  to  produce  torque  with  the  flux  in  the  SS'  axis. 
As  a  consequence,  the  motor  either  speeds  up  or  slows  down  until  the 
current  in  TT'  is  reduced  to  that  value  which  with  the  flux  in  the  SS' 
axis  will  give  sufficient  torque  to  overcome  the  torque  due  to  the  load, 
the  friction  and  losses  of  the  machine. 

The  action  while  similar  is  not  exactly  the  same  as  that  in  the  shunt- 
wound  direct-current  motor.  If,  for  example,  the  motor  speeds  up, 
the  speed  e.m.f.  in  the  SS'  axis  is  increased.  This  immediately  causes 
a  greater  flux  through  the  SS'  axis  to  provide  the  correspondingly 
greater  transformer  e.m.f.  in  this  circuit.  This,  in  turn,  means  that 
the  speed  of  the  motor  does  not  have  to  increase  so  much  as  would  other- 
wise be  the  case  in  order  that  the  increased  speed  voltage  in  the  TT' 


216 


THE  INDUCTION  MOTOR 


axis  may  be  generated.     Hence,  the  increase  in  speed  in  not  in  the 

A  similar  argument  will 


ratio  of  — ^ —  but  in  the  ratio 


FIG.    116. — Speed  Control  of  Single- 
phase  Motor  by  Armature  Resistance. 


apply  in  the  case  of  reduced  speed,  the  speed  field  in  this  case  being 

weakened.    The  speed  may  also  be  lowered,  as  shown  in  Fig.  116,  by 

inserting  resistance  in  the  TT' 
circuit.  This  has  the  effect  of 
weakening  the  current  in  this 
circuit  until  the  motor  slows 
down  sufficiently  to  make  the 
difference  of  the  two  e.m.fs. 
great  enough  to  establish  the 
requisite  current.  Using  this 
method,  the  speed  for  varying 
loads  will  not  be  constant,  and 
there  will  be  a  large  loss  of 
power  in  the  regulating  resist- 
ance. The  arrangement  is  simi- 
lar to  that  of  a  shunt-wound 

direct-current   motor   the  speed   of   which  is  controlled   by  adjusting 

the  resistance  of  the  armature  circuit.     Obviously,   only  a  decrease   of 

speed  can  be  obtained  by  this  means. 

In  Figs.  117  and  118  are  shown  two  of  the  many  possible  methods 

of  speed  control  by  field  variation.     If  an  e.m.f.  in  phase  with  the  line 

voltage  be  introduced    into  the 

brushes  SS',     it  will  have  little 

or  no  effect  on  the  speed.    This 

is  in  fact  exactly  what  we  do  to 

improve    the     power-factor    by 

compensation,  and  as  was  shown 

in  the  discussion   of   compensa- 

tion, the  speed  is  not  materially 

changed. 

In  the  connection   as    made 

in  Fig.  117,  if  the  windings  are 

so  connected    that   the  ampere- 

turns  Of  the  rotor  oppose   those 
' 


A 


FlG"  "7--Speed  Adjustment   of  Single- 
phase  Motor  by  Field  Control. 


,     ,  .,    .       ,,  j        . 

of  the   coil  in   the    speed  axis, 

the    flux   along   this   axis    will   be    lessened.      If    connected  in   the 
opposite   manner  it  will    of  course   be    strengthened.     If  the  former. 


THE  SINGLE-PHASE   INDUCTION  REPULSION  MOTOR        217 

the  speed  e.m.f.  in  the  transformer  axis  will  be  lessened  and  the  motor 
will  operate  at  a  higher  speed  in  order  to  make  this  e.m.f.  approximately 
equal  to  the  transformer  e.m.f.  If  on  the  other  hand  the  flux  be  increased 
the  motor  of  course  operates  at  a  higher  speed. 

Another  way  of  obtaining  the  same  effect  is  shown  in  Fig.  118. 
A  reactance  is  connected  in  the  circuit  of  the  speed  brushes,  and  taps 
are  brought  out  so  that  the  value  of  the  reactance  can  be  readily  varied. 
The  action  may  be  looked  at  in  two  ways.  We  may  consider  that  we  are 
inserting  an  e.m.f.  in  the  circuit  of  the  speed  brushes.  This  e.m.f. 


FIG.  1 1 8. — Single-phase  Motor  with 

Speed  Control  by  Means  of 
Reactance  in  the  Field  Circuit. 


FIG.  119. — Connections  of  Wagner 
Type  B.K.  Motor. 


is  the  back  e.m.f.  due  to  the  reactance.  It  is,  of  course,  90  degrees 
in  phase  from  the  current  or  in  phase  with  the  speed  and  transformer 
e.m.fs.  in  the  axis.  This  would,  as  we  have  shown,  have  the  effect  of 
reducing  the  flux  in  this  axis  and  consequently  of  increasing  the  speed 
of  the  motor. 

The  other  way  of  looking  at  the  matter  is  to  consider  that  we  have 
transfered  a  portion  of  the  flux  to  a  location  where  it  is  still  available 
for  generating  the  transformer  e.m.f.  in  the  speed  axis,  but  is  not  avail- 
able for  generating  speed  e.m.f.  In  order  therefore  to  generate  the 
proper  speed  e.m.f.  in  the  TT'  axis  due  to  cutting  the  flux  in  the  SS' 
axis,  the  motor  will  be  obliged  to  rotate  at  a  greater  speed.  This 


218  THE  INDUCTION   MOTOR 

method  is  obviously  available  only  for  increasing  the  speed,  and  not 
for  reducing  it. 

Another  motor  operating  on  the  same  general  principles  is  the 
Wagner  type  BK.  This  motor  is  provided  with  a  commutated  winding 
of  the  same  nature  as  a  direct-current  winding,  but  has  in  addition  a 
squirrel-cage  winding  of  the  usual  type.  The  electrical  connections 
are  shown  in  Fig.  119;  and  a  section  of  a  slot  showing  both  the  com- 
mutated winding  and  the  squirrel-cage  is  given  in  Fig.  120. 

During  starting,  the  switch  "  9  "  is  open,  and  the  winding  "  2  " 
carries  no  current.  Disregarding  for  the  moment  the  squirrel-cage, 
the  connections  of  the  motor  are  the  same  as  those  of  the  so-called 
compensated  repulsion  motor.  Such  motors  are  used  to  some  extent 
as  single -phase  railway  motors,  and  of  course  have  series  characteristics. 
The  motor  therefore  starts  with  excellent  torque.  The  torque  due  to 
this  winding,  however,  decreases  rapidly  as  the  speed  increases,  as  is 
the  case  with  all  series  motors. 

Retaining  Wedge. 

mmutated  Winding. 
Magnetic  Separator. 

ulrrel  Cage  Winding. 

FIG.  120.— Slot  of  Wagner  Type  B.K  Motor- 
In  addition  to  this  torque  there  is  a  torque  due  to  the  squirrel-cage. 
This  latter  torque  is  zero  at  start,  rapidly  increases  as  the  speed  increases, 
reaching  a  maximum  at  about  4  or  5  per  cent  slip,  and  becomes  zero 
at  synchronism.  Above  synchronism  the  torque  due  to  the  squirrel- 
cage  reverses.  The  general  shape  of  this  curve  is  as  shown  in  Fig.  103. 
The  torque  curve  of  the  motor  is  the  resultant  of  these  two  curves. 
There  is  a  peak  in  the  curve  at  zero  speed,  a  slight  lowering  for  inter- 
mediate speeds,  and  another  and  slightly  higher  peak  at  about  90  per 
cent  of  synchronous  speed.  With  a  starting  torque  of  160  per  cent,  the 
minimum  torque  during  acceleration  is  stated  to  be  133  per  cent. 

The  squirrel-cage  winding  is  intended  to  be  comparatively  inactive 
at  low  speeds,  and  to  carry  the  principal  part  of  the  current  at  speeds 
near  synchronism.  This  object  is  accomplished  by  placing  the  squirrel- 
cage  winding  in  the  bottom  of  the  slots  and  using  the  steel  bars  shown 
in  Fig.  120,  between  the  two  windings.  The  use  of  these  steel  bars 
causes  the  local  leakage  reactance  of  the  squirrel-cage  to  be  high.  At 
start,  the  frequency  of  the  currents  in  the  squirrel-cage  is  the  same  as 


THE  SINGLE-PHASE  INDUCTION  REPULSION  MOTOR        219 

the  line  frequency.  On  account  of  the  high  reactance,  the  secondary 
current  produced  is  small,  and  the  commutated  winding  carries  the 
larger  part  of  the  current. 

When  the  motor  has  nearly  reached  synchronism,  a  centrifugal 
switch  operates  to  close  the  circuit  through  the  switch  "  9."  The 
motor  then  operates  as  far  as  the  commutated  winding  is  concerned 
in  exactly  the  same  manner  as  the  ordinary  compensated  single -phase 
commutator  type  motor.  It  will  be  noted  that  with  the  switch  "  9  " 
closed,  the  winding  "  2  "  becomes  a  part  of  the  stator  winding.  The 
squirrel-cage  winding,  however,  takes  its  share  of  the  current  and  the 
result  is  that  the  slip  of  the  motor  under  load  is  very  small. 

The  squirrel-cage  is  also  of  advantage  in  that  is  prevents  the  pos- 
sibility of  the  motor  running  away.  Without  it,  if  one  of  the  brushes 
"  7  "  or  "  8  "  should  cease  to  make  contact  with  the  commutator  and 
the  load  on  the  motor  were  light,  the  motor  might  speed  up  greatly 
and  damage  itself.  With  the  squirrel-cage  winding  this  is  impossible. 
Even  though  all  of  the  brushes  should  cease  to  make  contact  with  the 
commutator,  the  motor  would  operate  as  before,  and  no  change  would 
be  apparent  to  the  eye.  The  capacity  would  however  be  reduced,  and 
the  power-factor  lowered.  The  motor  would,  however,  continue  to 
operate  indefinitely  without  injury. 


INDEX 


PAGE 

Adams,  Comfort  A 114 

method  for  determining  leakage  coefficient 114 

Armature  of  commutator-type  motor 87 

Auto  starter 42 

Auto-transformer 43 

Behn-Eschenberg,  Dr.  Hans 114 

method  for  determining  leakage  coefficient 114 

Behrend,  B.  A 115 

method  for  determining  leakage  coefficient 115 

Belt  leakage no 

Cascade  connection  of  induction  motors 77 

Circle  diagram 21 

exact  construction 27 

of  synchronous  motor 84 

Circuit  breakers,  use  of 56 

Commutator  type  of  induction  motor 82 

armature  of 87 

commutator  of 93 

connections  of 87 

current  in  armature  of 88 

impedance  of 89 

power  factor  of 86 

Concatenation,  connection  of  motors  in 77 

Connections  for  changing  number  of  poles 82 

Controller,  secondary 75 

Copper  loss 136 

Current : 

distribution  of  in  stator 4 

magnetizing 100 

rotor .  , 6 

three-phase 3 

Design  of  50  horse-power  motor 156 

Distribution  of  current  in  stator 4 

Distribution  of  magnetizing  current 103 

Distributed  windings 96 

221 


222  INDEX 

PAGE 

E.m.f .,  generation  of 7 

End-connection  leakage no 

End  rings,  size  of 161 

Estimation  of  heating 141 

Flux  wave,  shape  of 8 

influence  of  rotor  current  upon 9 

Fractional  pitch  windings 98 

effect  upon  starting  torque 150 

effect  upon  weight  of  copper  required 147 

Frames: 

enclosed  type,  General  Electric 182 

heavy  duty  type,  Westinghouse 180 

ordinary  type,  Fairbanks-Morse 178 

pedestal  type,  General  Electric 181 

riveted  type,  General  Electric 180 

Frequency,  influence  upon  performance 123 

Generators,  induction 60 

connections  of 62 

excitation  of 61 

power  factor  of 63 

uses  of 65 

Hansen,  I.  E 141 

Heating,  estimation  of 141 

Hobart,  H.  M 116 

method  for  determining  leakage  coefficient 116 

Impedance  of  armature  of  commutator-type  induction  motor 89 

Induction  generator 60 

connections  of 65 

excitation  of 62 

power  factor  of 63 

uses  of 6 1 

Iron  loss  in  rotor 139 

Leakage,  magnetic 108 

belt no 

coefficient 108,  113 

determination  of in 

end  connection no 

fluxes 109 

tooth no 

zig-zag no 

Losses: 

copper 136 

iron 137 

iron  in  rotor ' 139 


INDEX  223 

PAGE 

McAllister.  A.  S 116 

method  for  determining  leakage  coefficient 116 

Magnetizing  current 100 

distribution  of 103 

in  two-  and  three-phase  motors 106 

Motors,  single-phase 194 

See  also  Single-phase  motors. 

Overload,  protection  against : 56 

Phase  converter 67 

connections  of 69 

regulation  of 70 

Power  factor 26 

diameter  of  rotor  for  best 131 

maximum  of  induction  motor 26 

of  commutator-type  induction  motor 86 

of  induction  generator 63 

Resistance  starters .' 46 

Rotor: 

best  diameter  of 128 

current  in 6 

iron  loss  of 139 

Rotors,  types  of: 

Crocker-Wheeler 185 

Fairbanks-Morse 183 

General  Electric 184 

Westinghouse 182,  185 

Secondary  controller 75 

Secondary  starter 76 

Short  pitch  windings 98 

Single-phase  motors: 

commutator  type 201 

compensation  of 211 

current  distribution  in 198 

field  in 195 

General  Electric,  type  R.I 213 

repulsion   motor 209 

slip  in 204 

speed  control  of 214 

speed  torque  curves  of 206 

torque  in 204 

Wagner  type  B.K 217 

with  high  power  factor 211 


224  INDEX 


of  induction  generator  ............................................  62 

of  induction  motor  ...............................................  24 

Slots,  effect  of  upon  power  factor  .......................................  127 

open  or  closed  ...................................................  I26 

Special  types  of  motors: 

Burke  induction  motor  ............................................  1  73 

General  Electric,  mill  type  ........................................  1  76 

General  Electric,  type  L  ..........................................  167 

squirrel-cage,  crane  motors  ........................................  174 

Wagner,  type  B.W  ................................................  168 

Westinghouse,  mill  type  ..........................................  177 

Speed  torque  curves  of  induction  motor  .................................  73 

Starters  ...........................................  .  ..................  41 

adjustable  .......................................................  52 

auto-transformer  type  ............................................  42 

carbon-block  type  ................................................  53 

comparison  of  auto-transformer  and  resistance  types  ..................  50 

secondary  .......................................................  76 

Starting  conditions  ...................................................  14 

Starting  connections  for  starting  several  motors  ..........................  49 

Starting  current,  curve  of  .............................................  55 

Starting  torque  ......................................................  30 

comparative  of  various  motors  .....................................  30 

comparative  value  of  in  squirrel-cage  and  wound-rotor  machines  .......  36 

effect  of  iron  losses  upon  ..........................................  38 

fluctuation  of  ....................................................  39 

influence  of  leakage  upon  .  .........................................  37 

practical  values  of  ................................................  33 

Synchronous  motor,  circle  diagram  of  ...................................  84 

Theory,  elementary,  of  induction  motor  ................................  i 

Three-phase  current  ..................................................  3 

Tooth  leakage  ..........................................  .............  no 

Torque  .............................................................  24 

production  of  ....................................................  7 

starting  ......................................................  30,  164 

torque  and  speed  curves  of  wound-rotor  induction  motor  ..............  73 

Transformer,  ideal  ...................................................  18 

Vector  relations  of  current  and  e.m.f  ...................................  .  .   10 

Ventilation,  effect  upon  heating  ........................................  143 

Windings  ............................................................  151 

Windings,  various  types  of: 

basket  ..........................................................  154 

concentric  .......................................................  iSS 

delta  ...........................................................  iS2 


INDEX  225 

Windings,  various  types  of:  PAGE 

diamond  coils 154 

distributed 96 

full  overlapping 154 

pole  connections 152 

short  pitch 98 

star 152 

Zig-zag  leakage no 


TK     Bailey  -  The  induction  motor. 
2785 

B1&  UNIVERSITY  OF  CALIFORNIA  LIBRARY 

Los  Angeles 
This  book  is  DUE  on  the  last  date  stamped  below. 


2  %  1962 


NdV'9"  ~  ttCfc 
DEC1&18W 

JAN     61885 

3W     619 
NOV  2  9  1971 

DEC  1  3  1971 


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Form  L9-25m-3,'6HB8165s4)444 


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